The Extended Uncertainty Principle from a Projector-Valued Measurement Perspective

TL;DR

This paper operationally revisits the Extended Uncertainty Principle by deriving sharp bounds on momentum spread and size constraints using apparatus-defined measures, extending to curved spaces and suggesting experimental tests.

Contribution

It provides a rigorous, apparatus-based derivation of the EUP bounds in flat and curved spaces, connecting geometric effects to operational measurements.

Findings

Derived sharp lower bounds on momentum and size product in 1D.

Extended bounds to 2D and 3D curved spaces with geometric interpretation.

Proposed experimental tests in diffraction, cold-atom, and optomechanics.

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.