Total momentum and other Noether charges for particles interacting in a quantum spacetime

TL;DR

This work tackles the fate of Noether charges for interacting particles in a quantum-spacetime with deformed relativistic symmetries, focusing on spatial 2D κ-Minkowski as a tractable toy model. Using a Hamiltonian, first-quantized approach, it derives deformed Noether charges and shows that total charges depend sensitively on how interactions are implemented, not only on free-particle kinematics. The authors demonstrate that three consistent charge-construction schemes emerge—proper-dS, κ-coproduct, and a symmetrized-dS variant—each selected by specific Hamiltonians and sometimes requiring additional higher-order interaction terms; symmetrization can fail to yield conserved charges depending on the composition law. The results highlight that constructive derivations of conserved charges are essential for phenomenology in quantum spacetimes and likely generalize beyond κ-Minkowski to other deformed-symmetry scenarios.

Abstract

There has been strong interest in the fate of relativistic symmetries in some quantum spacetimes, also because of its possible relevance for high-precision experimental tests of relativistic properties. However, the main technical results obtained so far concern the description of suitably deformed relativistic-symmetry transformation rules, whereas the properties of the associated Noether charges, which are crucial for the phenomenology, are still poorly understood. We here tackle this problem focusing on first-quantized particles described within a Hamiltonian framework and using as toy model the so-called ``spatial kappa-Minkowski noncommutative spacetime'', where all the relevant conceptual challenges are present but, as here shown, in technically manageable fashion. We derive the Noether charges, including the much-debated total-momentum charges, and we expose a strong link between the properties of these Noether charges and the structure of the laws of interaction among particles.