Noncommutative spaces of worldlines

Abstract

The space of time-like geodesics on Minkowski spacetime is constructed as a coset space of the Poincaré group in (3+1) dimensions with respect to the stabilizer of a worldline. When this homogeneous space is endowed with a Poisson homogeneous structure compatible with a given Poisson-Lie Poincaré group, the quantization of this Poisson bracket gives rise to a noncommutative space of worldlines with quantum group invariance. As an oustanding example, the Poisson homogeneous space of worldlines coming from the κ-Poincaré deformation is explicitly constructed, and shown to define a symplectic structure on the space of worldlines. Therefore, the quantum space of κ-Poincaré worldlines is just the direct product of three Heisenberg-Weyl algebras in which the parameter κ −1 plays the very same role as the Planck constant in quantum mechanics. In this way, noncommutative spaces of worldlines are shown to provide a new suitable and fully explicit arena for the description of quantum observers with quantum group symmetry.

The κ-(A)dS noncommutative spacetime

“…Indeed, the consequences of making use of the κ-(A)dS noncommutative spacetime (2) from different quantum gravity perspectives have to be explored promptly by following similar approaches to the ones used so far for the κ-Minkowski spacetime (see, for instance, [22]- [39]). Work on this line is currently in progress and will be presented elsewhere.…”

“…Moreover, it is straightforward to check that the Poisson brackets (36) arise in the Sklyanin bracket just from the J 1 ∧ J 2 term of the r-matrix (30). This explains why the Poisson algebra (39) is naturally linked to the semiclassical limit of the quantum SU(2) subgroup of the κ-(A)dS deformation, albeit realized on the 3-space coordinates. In this respect, we recall that the su(2) subalgebra generated by {J 1 , J 2 , J 3 } becomes a quantum su(2) subalgebra when the full quantum deformation is constructed [45], a fact that can already be envisaged from the cocommutator (34) where the su(2) generators define a sub-Lie bialgebra.…”

“…Thus space coordinates become noncommutative, while at first-order in η the time-space sector is kept invariant with respect to the κ-Minkowski case. Moreover, the space-space brackets (39) are just a subalgebra of the quantum SU(2) group. Now, the quantum κ-(A)dS spacetime for any order in η should be obtained as the quantization of the full Poisson algebra (35)- (36), which is by no means a trivial task due to the noncommutativity of the space coordinates given by (36).…”

“…Since then, the κ-Minkowski spacetime has provided a privileged benchmark for the implementation of a number of models aiming to describe different features of quantum geometry at the Planck scale and their connections with ongoing phenomenological proposals. Without pretending to be exhaustive, κ-Minkowski spacetime has been studied in relation with wave propagation on noncommutative spacetimes [22], Deformed Special Relativity features [23], dispersion relations [24,25,26], relative locality phenomena [27], curved momentum spaces and phase spaces [28,29], noncommutative differential calculi [30,31], star products [32], noncommutative field theory [33,34,35], representation theory [36,37], light cones [38] and noncommutative spaces of worldlines [39].…”

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).…”