The more things change the more they stay the same: Minimum lengths with unmodified uncertainty principle and dispersion relation

TL;DR

This paper develops a minimal length quantum gravity model by modifying operators while preserving core quantum mechanics and relativity structures, potentially addressing cosmic ray puzzles without violating Lorentz symmetry.

Contribution

It introduces a minimal length model that maintains standard quantum and relativistic relations, avoiding Lorentz violation constraints while addressing high-energy cosmic ray phenomena.

Findings

Retains Lorentz symmetry in the minimal length model

Potential to explain ultra high energy cosmic ray observations

Avoids constraints from photon dispersion experiments

Abstract

Broad arguments indicate that quantum gravity should have a minimal length scale. In this essay we construct a minimum length model by generalizing the time-position and energy-momentum operators while keeping much of the structure of quantum mechanics and relativity intact: the standard position-momentum commutator, the special relativistic time-position, and energy-momentum relationships all remain the same. Since the time-position and energy-momentum relationships for the modified operators remains the same, we retain a form of Lorentz symmetry. This avoids the constraints on these theories coming from lack of photon dispersion while holding the potential to address the Greisen-Zatsepin-Kuzmin (GZK) puzzle of ultra high energy cosmic rays.