  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 $\kappa$-Minkowski type
non-commutative (quantum) space-time, where the Poincare algebra, on its own,
remains undeformed, but in order to retain the structure of the space-time
non-commutative (NC) algebra, action 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.
