Localization and reference frames in κ -Minkowski spacetime

Abstract

We study the limits to the localizability of events and reference frames in the κ-Minkowski quantum spacetime. Our main tool will be a representation of the κ-Minkowski commutation relations between coordinates, and the operator and measurement theory borrowed from ordinary quantum mechanics. Spacetime coordinates are described by operators on a Hilbert space, and a complete set of commuting observables cannot contain the radial coordinate and time at the same time. The transformation between the complete sets turns out to be the Mellin transform, which allows us to discuss the localizability properties of states both in space and time. We then discuss the transformation rules between inertial observers, which are described by the quantum κ-Poincaré group. These too are subject to limitations in the localizability of states, which impose further restrictions on the ability of an observer to localize events defined in a different observer's reference

Fuzzy worldlines with κ-Poincaré symmetries

“…In fact, computed on states such that cosh η = 1, the right-hand side of (4.17) is zero. In this subsection we show that these states can be defined as limits of normalized states, as was done in [48,49] for the states localized at the spatial origin in κ-Minkowski spacetime. In this way, we are able to define a state in the origin of the space of worldlines, corresponding to the worldline w 0 , that can be used as a sharp reference to compute the impact parameter (2.9) of a quantum worldline w with respect to it.…”

Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

“…Among other issues and within the vast literature, let us mention that κ-Minkowski space along with the κ-Poincaré algebra have been studied in relation to noncommutative differential calculi [72,73], wave propagation on noncommutative spacetimes [74], deformed or doubly special relativity at the Planck scale [75][76][77][78][79][80], noncommutative field theory [81][82][83], representation theory on Hilbert spaces [84,85], generalized κ-Minkowski spacetimes through twisted κ-Poincaré deformations [86,87], deformed dispersion relations [88][89][90], curved momentum spaces [91][92][93][94][95], relative locality phenomena [96], star products [97], deformed phase spaces [98], noncommutative spaces of worldlines [99,100] and light cones [101] (in all cases see the references therein).…”

“…(1) ω , without taking into account other spaces. Along this paper, a CK geometry will be understood as the full set of the four homogeneous spaces (85).…”

“…It is possible to construct other 4D and 6D symmetric homogeneous spaces from the CK group SO ω (5), which, depending on each particular CK geometry, could be different from the four above ones (85). In particular, any composition of the automorphisms Θ (m) (33), which form a basis for the Z ⊗4 2 -grading, gives rise to another automorphism that provides another Cartan decomposition, like (83), and, from it, the corresponding coset space can be constructed.…”

“…and fulfilling(84). As we already performed in Section 3.3 for so(5) * , we define the first-order noncommutative CK spaces by S is the first-order in the quantum coordinates of the complete noncommutative space associated with the homogeneous CK space S (m) ω(85). We display, in Table3, the defining commutation relations for the four noncommutative CK spaces along with the Lie brackets among h(m) ⊥,ω and t (m) ⊥,ω .…”