Fate of $κ$-Minkowski space-time in non-relativistic (Galilean) and ultra-relativistic (Carrollian) regimes

TL;DR

The work investigates the fate of κ-Minkowski space-time under non-relativistic and ultra-relativistic limits by applying Wigner–Inönü contractions to the Poincaré algebra and to κ-deformed phase spaces. A central outcome is the branching of the non-commutative parameter a^μ: in the Galilean limit only spatial components survive (a^0=0), producing a fundamental length scale and a flat 3D momentum subspace, while in the Carrollian limit only the temporal component survives (a^i=0) giving a fundamental time scale and a momentum space that is effectively one-dimensional along energy. The authors construct explicit Bopp maps for both κ-Galilean and κ-Carrollian coordinates, derive the corresponding deformed phase-space algebras, and compute curved momentum-space metrics and geodesic distances that yield deformed dispersion relations. They find that the deformed Casimirs share the same functional form in both limits but depend on different momentum-space arguments, reflecting the distinct degeneracies of the limiting geometries. The results illuminate how non-commutativity and quantum-gravity effects can persist in extreme kinematic regimes and suggest avenues for multi-particle extensions and potential condensed-matter analogs via curved momentum-space scenarios.

Abstract

Here, we present an algebraic and kinematical analysis of non-commutative $κ$-Minkowski spaces within Galilean (non-relativistic) and Carrollian (ultra-relativistic) regimes. Utilizing the theory of Wigner-Inönu contractions, we begin with a brief review of how one can apply these contractions to the well-known Poincaré algebra, yielding the corresponding Galilean (both massive and mass-less) and Carrollian algebras as $c \to \infty$ and $c\to 0$, respectively. Subsequently, we methodically apply these contractions to non-commutative $κ$-deformed spaces, revealing compelling insights into the interplay among the non-commutative parameters $a^μ$ (with $|a^ν|$ being of the order of Planck length scale) and the speed of light $c$ as it approaches both infinity and zero. Our exploration predicts a sort of "branching" of the non-commutative parameters $a^μ$, leading to the emergence of a novel length scale and time scale in either limit. Furthermore, our investigation extends to the examination of curved momentum spaces and their geodesic distances in appropriate subspaces of the $κ$-deformed Newtonian and Carrollian space-times. We finally delve into the study of their deformed dispersion relations, arising from these deformed geodesic distances, providing a comprehensive understanding of the nature of these space-times.