The Extended Uncertainty Principle from a Projector-Valued Measurement Perspective

Abstract

We revisit the Extended Uncertainty Principle (EUP) from an operational viewpoint, replacing wavefunction-based widths with apparatus-defined position constraints such as a finite slit of width $\Delta x$ or a geodesic ball of radius $R$. Using Hermitian momentum operators consistent with the EUP algebra, we prove a sharp lower bound on the product of momentum spread and preparation size in one dimension and show that it reduces smoothly to the standard quantum limit as the deformation vanishes. We then extend the construction to the dimensions two and three on spaces of constant curvature and obtain the corresponding bound for spherical confinement, clarifying its geometric meaning via an isometry to $S^2$ and $S^3$. The framework links curvature-scale effects to operational momentum floors and suggests concrete tests in diffraction, cold-atom, and optomechanical settings.

arXiv comment15 pages

References in corpus (19)

• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• Quantum gravity and minimum length
• Generalized Uncertainty Principle in Quantum Gravity from Micro-Black Hole Gedanken Experiment
• The Generalized Uncertainty Principle and Black Hole Remnants
• Loop Quantum Gravity
• Universality of Quantum Gravity Corrections
• Discreteness of Space from GUP II: Relativistic Wave Equations
• The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length
• (Anti-)de Sitter Black Hole Thermodynamics and the Generalized Uncertainty Principle
• Extended uncertainty principle and the geometry of (anti)-de Sitter space
• A closer look at the uncertainty relation of position and momentum
• Uncertainty principle on 3-dimensional manifolds of constant curvature
• A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas
• Generalized Extended Momentum Operator
• Generalized Extended Uncertainty Principles, Liouville theorem and density of states: Snyder-de Sitter and Yang models
• Extended Uncertainty Principle: A Deeper Insight into the Hubble Tension?
• Scattering and absorption by extra-dimensional black holes with GUP
• On momentum operators given by Killing vectors whose integral curves are geodesics
• On the experimental verification of the uncertainty principle of position and momentum