Physically motivated uncertainty relations at the Planck length for an emergent non commutative spacetime

TL;DR

This paper derives new space-time uncertainty relations at the Planck scale from quantum mechanics and general relativity, indicating minimal volumes rather than lengths and establishing a maximum energy density.

Contribution

It introduces a novel derivation of space-time uncertainty relations that challenge previous claims of minimal length, emphasizing minimal volumes and energy density bounds.

Findings

No minimal length uncertainty, only minimal volume and four-volume uncertainties

Derived a maximal bound for energy density at the Planck scale

Proposed commutator structures consistent with the new uncertainty relations

Abstract

We derive new space-time uncertainty relations (STUR) at the fundamental Planck length $L_P$ from quantum mechanics and general relativity (GR), both in flat and curved backgrounds. Contrary to claims present in the literature, our approach suggests that no minimal uncertainty appears for lengths, but instead for minimal space and four-volumes. Moreover, we derive a maximal absolute value for the energy density. Finally, some considerations on possible commutators among quantum operators implying our STUR are done.