Two one-parameter families of twists providing $\kappa$-Minkowski $*$ product deformed spacetime are considered: Abelian and Jordanian. We compare the derivation of quantum Minkowski space from two perspectives. The first one is the Hopf module algebra point of view, which is strictly related with Drinfeld’s twisting tensor technique. The other one relies on an appropriate extension of “deformed realizations” of nondeformed Lorentz algebra by the quantum Minkowski algebra. This extension turns out to be de Sitter Lie algebra. We show the way both approaches are related. The second path allows us to calculate deformed dispersion relations for toy models ensuing from different twist parameters. In the Abelian case, one recovers $\kappa$-Poincaré dispersion relations having numerous applications in doubly special relativity. Jordanian twists provide a new type of dispersion relations which in the minimal case (related to Weyl-Poincaré algebra) takes an energy-dependent linear mass deformation form.

Several issues concerning quantum kappa-Poincare algebra are discussed and reconsidered here. We propose two different formulations of kappa-Poincare quantum algebra. Firstly we present a complete Hopf algebra formulae of kappa-Poincare in classical Poincare basis. Further by adding one extra generator, which modifies the classical structure of Poincare algebra, we eliminate non polynomial functions in the kappa-parameter. Hilbert space representations of such algebras make Doubly Special Relativity (DSR) similar to the Stueckelberg's version of (proper-time) relativistic Quantum Mechanics.

We compare two versions of deformed dispersion relations (energy vs. momenta and momenta vs. energy) and the corresponding time delay up to the second-order accuracy in the quantum gravity scale (deformation parameter). A general framework describing modified dispersion relations and time delay with respect to different noncommutative κ-Minkowski spacetime realizations is firstly proposed here and it covers all the cases introduced in the literature. It is shown that some of the realizations provide certain bounds on quadratic corrections, i.e. on quantum gravity scale, but it is not excluded in our framework that the quantum gravity scale is the Planck scale. We also show how the coefficients in the dispersion relations can be obtained through a multiparameter fit of the gamma-ray burst (GRB) data.

We discuss a quantum deformation of the Green-Schwarz superstring on flat space, arising as a contraction limit of the corresponding deformation of ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$. This contraction limit turns out to be equivalent to a previously studied limit that yields the so-called mirror model—the model obtained from the light cone gauge fixed ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ string by a double Wick rotation. Reversing this logic, the ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ superstring is the double Wick rotation of a quantum deformation of the flat space superstring. This quantum deformed flat space string realizes symmetries of the timelike $\kappa$-Poincaré type and is T dual to ${\mathrm{dS}}_5\times {\mathrm{H}}^5$, indicating interesting relations between symmetry algebras under T duality. Our results directly extend to ${\mathrm{AdS}}_2\times {\mathrm{S}}^2\times {\mathrm{T}}^6$ and ${\mathrm{AdS}}_3\times {\mathrm{S}}^3\times {\mathrm{T}}^4$ and beyond string theory to many (semi)symmetric space coset sigma models, such as a deformation of the four-dimensional Minkowski sigma model with timelike $\kappa$-Poincaré symmetry. We also discuss possible null and spacelike deformations.

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of κ-Minkowski spacetime. The corresponding quantum Poincaré-Weyl Lie algebra of in-finitesimal translations, rotations and dilatations is obtained. The d’Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.