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

TL;DR

This paper derives the path integral action for a particle in $5$-Minkowski spacetime, revealing a dissipative term due to the deformation parameter, and analyzes the harmonic oscillator and free particle cases to understand the effects of this noncommutative geometry.

Contribution

It introduces a novel path integral formulation for particles in $5$-Minkowski spacetime, including the effects of $5$-deformation on dynamics and propagators.

Findings

Derived the path integral action with a dissipative term from $5$-deformation.

Calculated the oscillator frequency in the deformed spacetime.

Explicitly computed the free particle propagator in $5$-Minkowski spacetime.

Abstract

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.