Observers and Momenta in κ-Minkowski space-time

We study the limits to the localizability and the role of observers in κ-Minkowski quantum spacetime. Inspired by Quantum Mechanincs, we develop an interpretations of the non-commutativity in coordinates and of the deformations of transformation between observers. Space-time coordinates are operators on a Hilbert space. The transformation between the complete sets is realized by Mellin transform. Transformation rules between inertial observers are described by the quantum κ-Poincaré group. There are restrictions on the observer possibility to localize events. We also discuss the geometry of the curved momentum space dual to k-Minkowski coordinates, which turns out to be not unique. It can have any signature: Euclidean, Lorentzian, and (+,+,-,-), as well as degenerate cases. For any choice of a four dimensional metric there is a quantum group of symmetries of κ-Minkowski preserving it. We associate a momentum space to each nondegenerate choice of such metric. These momentum spaces are all maximally symmetric, and the isotropy subgroup of their isometries coincides with the homogeneous part of the quantum group. We also discuss the degenerate cases.