Nonlinear transforms of momenta and Planck scale limit

TL;DR

This paper introduces a nonlinear transformation of Poincaré momenta that imposes a Planck scale limit, preserving key invariants and symmetries, with implications for relativistic kinematics and algebraic structures.

Contribution

It proposes a novel nonlinear transformation of Poincaré generators that introduces a Planck scale limit while maintaining invariants and symmetries, extending to de Sitter and supersymmetric algebras.

Findings

Transformed momenta have a finite upper energy limit at the Planck scale.

The invariant mass squared remains conserved under the transformation.

The transformation preserves the speed of light as an absolute velocity scale.

Abstract

Starting with the generators of the Poincar\'e group for arbitrary mass (m) and spin (s) a nonunitary transformation is implemented to obtain momenta with an absolute Planck scale limit. In the rest frame (for $m>0$) the transformed energy coincides with the standard one, both being $m$. As the latter tends to infinity under Lorentz transformations the former tends to a finite upper limit $m\coth(lm) = l^{-1}+ O(l)$ where $l$ is the Planck length and the mass-dependent nonleading terms vanish exactly for zero rest mass.The invariant $m^{2}$ is conserved for the transformed momenta. The speed of light continues to be the absolute scale for velocities. We study various aspects of the kinematics in which two absolute scales have been introduced in this specific fashion. Precession of polarization and transformed position operators are among them. A deformation of the Poincar\'e algebra to the SO(4,1) deSitter one permits the implementation of our transformation in the latter case. A supersymmetric extension of the Poincar\'e algebra is also studied in this context.