Path integral action of a particle in $\kappa$-Minkowski spacetime

In this letter, we derive the path integral action of a particle in $\kappa$-Minkowski spacetime. The equation of motion for an arbitrary potential due to the $\kappa$-deformation of the Minkowski spacetime is then obtained. The action contains a dissipative term which owes its origin to the $\kappa$-Minkowski deformation parameter $a$. We take the example of the harmonic oscillator and obtain the frequency of oscillations in the path integral approach as well as operator approach upto the first order in the deformation parameter $a$. For studying this, we start with the $\kappa$-deformed dispersion relation which is invariant under the undeformed $\kappa$-Poincar$\acute{e}$ algebra and take the non-relativistic limit of the $\kappa$-deformed dispersion relation to find the Hamiltonian. The propagator for the free particle in the $\kappa$-Minkowski spacetime is also computed explicitly. In the limit, $a\rightarrow 0$, the commutative results are recovered.