How the Extended Uncertainty Principle

Roy B. Leipnik

Steven Matthysse

Abderrahmane Benhadjira, Abdelhakim Benkrane, Omar Bentouila +2 more

Abstract In this study, an analytical investigation was conducted to assess the effects of the extended uncertainty principle (EUP) on a Bose–Einstein condensate (BEC) described by the deformed one-dimensional Gross–Pitaevskii equation (GPE). Analytical solutions were derived for null potential while we used variational and numerical methods for a harmonic oscillator potential. The effects of EUP on stability, probability density, position, and momentum uncertainties of BEC are analyzed. The EUP is found to be applicable for the free dark soliton solution and in the presence of a harmonic potential within specific ranges of the deformation parameter α, while it is not valid for the free bright soliton solution.

A. Gamba

Matthew J. Lake, Marek Miller, Ray Ganardi +1 more

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 generalised uncertainty principle-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 regionσg> 0. This length scale is the standard deviation of the envelope function,g, that defines the positive operator valued measure elements. Similarly, an extended uncertainty principle-type relation emerges when the standard momentum operator is replaced by a positive operator valued measure that localises the wave packet to within a region σ̃g>0 in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by setting σg≃ℏG/c3 , the Planck length, and σ̃g≃ℏΛ/3 , where Λ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schrödinger and Heisenberg equations, remain unchanged. This demonstrates that generalised uncertainty principle and extended uncertainty principle 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.

Van Meter, Stephen

Project Description This document presents the Extended Heisenberg Uncertainty Principle (EHUP), a reformulation of quantum uncertainty that introduces time as an explicit conjugate parameter. The EHUP posits a tripartite relation, \Delta x \cdot \Delta p = K \cdot \Delta t where K = m_{0} c^{2} and m_{0} is defined as half the Planck mass. This framework unifies quantum mechanics and general relativity by linking spatial-momentum uncertainty to temporal resolution, thereby resolving long-standing inconsistencies in how physical systems evolve over time. The proposal demonstrates that temporal uncertainty is not a mathematical artifact but a physically necessary quantity governing the evolution of all dynamical systems. Preliminary evaluations against high-precision experimental domains—including GPS clock drift, KAGRA interferometry data, and Lunar Laser Ranging—show that incorporating \Delta t produces predictions that align more closely with observed data than the classical Heisenberg relation. If validated, the EHUP provides a coherent mathematical mechanism for time-dependent quantum behavior, gravitational decoherence, and information preservation in curved spacetime. It suggests that time is not merely a parameter but a quantized physical resource—one that constrains measurement, causality, and the evolution of complex systems.