$\kappa$-Minkowski Spacetime Through Exotic "Oscillator"

TL;DR

This paper introduces a covariant non-relativistic particle model representing $ppa$-Minkowski noncommutative spacetime, resulting in a novel 'Exotic Oscillator' with unique dynamical properties and preserved symmetries.

Contribution

It presents a new covariant particle model that captures $ppa$-Minkowski spacetime and reveals an Exotic Oscillator with distinctive energy-dependent frequency.

Findings

The Exotic Oscillator obeys a harmonic oscillator-like equation with frequency proportional to the square root of energy.

The model maintains discrete symmetries like parity and time reversal.

The phase space exhibits a singularity in momentum despite finite energy.

Abstract

We have proposed a generally covariant non-relativistic particle model that can represent the $\kappa$-Minkowski noncommutative spacetime. The idea is similar in spirit to the noncommutative particle coordinates in the lowest Landau level. Physically our model yields a novel type of dynamical system, (termed here as Exotic "Oscillator"), that obeys a Harmonic Oscillator like equation of motion with a {\it{frequency}} that is proportional to the square root of {\it{energy}}. On the other hand, the phase diagram does not reveal a closed structure since there is a singularity in the momentum even though energy remains finite. The generally covariant form is related to a generalization of the Snyder algebra in a specific gauge and yields the $\kappa $-Minkowski spacetime after a redefinition of the variables. Symmetry considerations are also briefly discussed in the Hamiltonian formulation. Regarding continuous symmetry, the angular momentum acts properly as the generator of rotation. Interestingly, both the discrete symmetries, parity and time reversal, remain intact in the $\kappa$-Minkowski spacetime.