Curved momentum space and finite Landau spectrum in $κ$-Minkowski spacetime

Abstract

We derive the $κ$-Poincaré Casimir from the de Sitter geometry of momentum space and employ it as the dynamical constraint governing charged particles within the framework of Poisson gauge theory. The resulting formalism is applied to scalar and spin-$1/2$ particles in a constant magnetic field. Within the semiclassical Poisson-gauge framework, exact energy spectra are obtained without expanding in the deformation parameter $1/κ$. On the positive-energy branch in bicrossproduct momentum coordinates, identifying these coordinates with the physical momenta entering the gauge coupling imposes an upper bound on their spatial magnitude. This bound leads to a finite number of admissible Landau-level indices and hence to a Highest Landau Level (HLL). In the fermionic case, the truncation is spin dependent for the factorization adopted here, resulting in a polarized HLL. Possible implications of this ultraviolet truncation and the limitations of the framework are briefly discussed.