Planck-scale soccer-ball problem: a case of mistaken identity

TL;DR

This paper clarifies that nonlinear momentum composition laws in quantum gravity do not necessarily imply macroscopic inconsistencies, distinguishing the roles of locality and translational invariance in total momentum calculations.

Contribution

It demonstrates that concerns about nonlinear momentum laws causing macroscopic issues are unfounded by differentiating their roles in locality and translational invariance.

Findings

Nonlinear momentum composition reflects locality, not total momentum.

Total momentum remains linearly additive due to translational invariance.

Misinterpretations stem from conflating the roles of locality and invariance.

Abstract

Over the last decade it has been found that nonlinear laws of composition of momenta are predicted by some alternative approaches to "real" 4D quantum gravity, and by all formulations of dimensionally-reduced (3D) quantum gravity coupled to matter. The possible relevance for rather different quantum-gravity models has motivated several studies, but this interest is being tempered by concerns that a nonlinear law of addition of momenta might inevitably produce a pathological description of the total momentum of a macroscopic body. I here show that such concerns are unjustified, finding that they are rooted in failure to appreciate the differences between two roles for laws composition of momentum in physics. Previous results relied exclusively on the role of a law of momentum composition in the description of spacetime locality. However, the notion of total momentum of a multi-particle system is not a manifestation of locality, but rather reflects translational invariance. By working within an illustrative example of quantum spacetime I show explicitly that spacetime locality is indeed reflected in a nonlinear law of composition of momenta, but translational invariance still results in an undeformed linear law of addition of momenta building up the total momentum of a multi-particle system.