Lorentz invariant nonzero minimal uncertainty in position and inhomogeneity of space at the Planck scale

TL;DR

This paper demonstrates that a nonzero minimal uncertainty in position measurements at the Planck scale is compatible with Lorentz invariance and explores its implications for space inhomogeneity and quantum algebra.

Contribution

It shows that minimal position uncertainty does not violate Lorentz invariance and determines the algebra between position and momentum at the Planck scale.

Findings

Minimal uncertainty in position is Lorentz invariant.

The algebra between position and momentum is fixed by Lorentz invariance.

Space becomes inhomogeneous at the Planck scale.

Abstract

The suspicion that the existence of a minimal uncertainty in position measurements violates Lorentz invariance seems unfounded. It is shown that the existence of such a nonzero minimal uncertainty in position is not only consistent with Lorentz invariance, but that the latter also fixes the algebra between position and momentum which gives rise to this minimal uncertainty. We also investigate how this algebra affects the underlying quantum mechanical structure, and why, at the Planck scale, space can no longer be considered homogeneous.