A New Approach to Generalised Uncertainty Relations

TL;DR

This paper proposes a novel model for generalized uncertainty relations that avoids modified commutation relations by introducing quantum background geometry, leading to a consistent extended uncertainty principle and implications for space quantization.

Contribution

It introduces a new approach where background geometry is quantized separately, avoiding issues of modified commutators and providing a consistent framework for extended uncertainty relations.

Findings

The model naturally produces the extended generalized uncertainty principle (EGUP).

It resolves issues like violation of the equivalence principle and the 'soccer ball' problem.

Space is quantized differently from matter, with fundamental quanta being fermions.

Abstract

We outline a new model in which generalised uncertainty relations are obtained without modified commutation relations. While existing models introduce modified phase space volumes for the canonical degrees of freedom, we introduce new degrees of freedom for the background geometry. The phase space is therefore enlarged but remains Euclidean. The spatial background is treated as a genuinely quantum object, with an associated state vector, and the model naturally gives rise to the extended generalised uncertainty principle (EGUP). Importantly, this approach solves (or rather, evades) well known problems associated with modified commutators, including violation of the equivalence principle, the `soccer ball' problem for multiparticle states, and the velocity dependence of the minimum length. However, it implies two radical conclusions. The first is that space must be quantised on a different scale to matter and the second is that the fundamental quanta of geometry are fermions. We explain how, in the context of the model, these do not contradict established results including the no go theorems for multiple quantisation constants, which still hold for species of material particles, and the spin-$2$ nature of gravitons.