The quest for a quantum gravity phenomenology has inspired a quantum notion of space-time, which motivates us to study the fate of the relativistic symmetries of a particular model of quantum space-time, as well as its intimate connection with the plausible emergent curved “physical momentum space”. We here focus on the problem of Poincare symmetry of κ-Minkowski type non-commutative (quantum) space-time, where the Poincare algebra, on its own, remains undeformed, but in order to preserve the structure of the space-time non-commutative (NC) algebra, the actions of the algebra generators on the operator-valued space-time manifold must be enveloping algebra valued that lives in entire phase space i.e. the cotangent bundle on the space-time manifold (at classical level). Further, we constructed a model for a spin-less relativistic massive particle enjoying the deformed Poincare symmetry, using the first order form of geometric Lagrangian, that satisfies a new deformed dispersion relation and explored a feasible regime of a future Quantum Gravity theory in which the momentum space becomes curved. In this scenario there is only a mass scale (Planck mass m_p), but no length scale. Finally, we relate the deformed mass shell to the geodesic distance in this curved momentum space, where the mass of the particle gets renormalized as a result of noncommutativity. We show, that under some circumstances, the Planck mass provides an upper bound for the observed renormalized mass.

We present an algebraic and kinematical analysis of non-commutative κ-Minkowski spaces within Galilean (non-relativistic) and Carrollian (ultra-relativistic) regimes. Utilizing the theory of Wigner-Inönu contractions, we begin with a brief review of how one can apply these contractions to the well-known Poincaré algebra, yielding the corresponding Galilean and Carrollian algebras as c → ∞ and c → 0, respectively. Subsequently, we methodically apply these contractions to non-commutative κ-deformed spaces, revealing compelling insights into the interplay among the non-commutative parameters a ^μ (with |a ^ν | being of the order of Planck length scale) and the speed of light c as it approaches both infinity and zero. Our exploration predicts a sort of “branching” of the non-commutative parameters a ^μ , leading to the emergence of a novel length scale and time scale in either limit. Furthermore, our investigation extends to the examination of curved momentum spaces and their geodesic distances in appropriate subspaces of the κ-deformed Newtonian and Carrollian space-times. We finally delve into the study of their deformed dispersion relations, arising from these deformed geodesic distances, providing a comprehensive understanding of the nature of these space-times.