A note on DSR-like approach to space-time

TL;DR

This paper explores the challenges of defining a space-time framework using a DSR-like approach, highlighting conflicts between Lorentz symmetry realization, invariant scales, and consistent translation group structures.

Contribution

It demonstrates that incorporating an invariant length scale in a DSR-inspired space-time model conflicts with maintaining a proper group structure of translations.

Findings

Nonlinear Lorentz realization conflicts with invariant length scale

Invariant scale prevents unambiguous definition of small distances

Group structure of translations cannot be maintained with invariant scale

Abstract

In this note we discuss the possibility to define a space-time with a DSR based approach. We show that the strategy of defining a non linear realization of the Lorentz symmetry with a consistent vector composition law cannot be reconciled with the extra request of an invariant length (time) scale. The latter request forces to abandon the group structure of the translations and leaves a space-time structure where points with relative distances smaller or equal to the invariant scale can not be unambiguously defined.