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The quest for a quantum gravity phenomenology has inspired a quantum notionof space-time, which motivates us to study the fate of the relativisticsymmetries of a particular model of quantum space-time, as well as its intimateconnection with the plausible emergent curved "physical momentum space". Wehere focus on the problem of Poincare symmetry of $\kappa$-Minkowski typenon-commutative (quantum) space-time, where the Poincare algebra, on its own,remains undeformed, but in order to retain the structure of the space-timenon-commutative (NC) algebra, action of the algebra generators on theoperator-valued space-time manifold must be enveloping algebra valued thatlives in entire phase space i.e. the cotangent bundle on the space-timemanifold (at classical level). Further, we constructed a model for a spin-lessrelativistic massive particle enjoying the deformed Poincare symmetry, usingthe first order form of geometric Lagrangian, that satisfies a new deformeddispersion relation and explored a feasible regime of a future Quantum Gravitytheory in which the momentum space becomes curved. In this scenario there isonly a mass scale (Planck mass $m_{p}$), but no length scale. Finally, werelate the deformed mass shell to the geodesic distance in this curved momentumspace, where the mass of the particle gets renormalized as a result ofnoncommutativity. We show, that under some circumstances, the Planck massprovides an upper bound for the observed renormalized mass.

Symmetries of $\kappa$ Minkowski space-time: A possibility of exotic momentum space geometry?