Lorentz invariance, 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 can coexist with Lorentz invariance and explores how this affects quantum structure and space homogeneity at the Planck scale.

Contribution

It shows that minimal position uncertainty is compatible with Lorentz invariance and determines the algebra between position and momentum, impacting quantum mechanics at the Planck scale.

Findings

Minimal uncertainty in position does not violate Lorentz invariance.

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

Space homogeneity breaks down 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.