Generalised Uncertainty Relations from Finite-Accuracy Measurements

TL;DR

This paper demonstrates that generalized uncertainty relations like GUP and EUP can be derived within standard quantum mechanics using finite-accuracy measurements, avoiding the need for modified commutation relations and their associated issues.

Contribution

It shows how GUP and EUP arise from POVMs representing finite measurement precision, without altering canonical quantum theory or its fundamental equations.

Findings

GUP and EUP can be derived without modified commutators.

Finite-accuracy measurements naturally lead to these uncertainty relations.

Standard quantum mechanics remains consistent without pathologies.

Abstract

In this short note we show how the Generalised Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP), two of the most common generalised uncertainty relations proposed in the quantum gravity literature, can be derived within the context of canonical quantum theory, without the need for modified commutation relations. A GUP-type relation naturally emerges when the standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises the quantum wave packet to within a spatial region $\sigma_g > 0$. This length scale is the standard deviation of the envelope function, $g$, that defines the POVM elements. Similarly, an EUP-type relation emerges when the standard momentum operator is replaced by a POVM that localises the wave packet to within a region $\tilde{\sigma}_g > 0$ in momentum space. The usual GUP and EUP are recovered by setting $\sigma_g \simeq \sqrt{\hbar G/c^3}$, the Planck length, and $\tilde{\sigma}_g \simeq \hbar\sqrt{\Lambda/3}$, where $\Lambda$ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schr{\" o}dinger and Heisenberg equations, remain unchanged. This demonstrates that GUP and EUP phenomenology can be obtained without modified commutators, which are known to lead to various pathologies, including violation of the equivalence principle, violation of Lorentz invariance in the relativistic limit, the reference frame-dependence of the `minimum' length, and the so-called soccer ball problem for multi-particle states.