On Majorization Uncertainty Relations in the Presence of a Minimal Length

Abstract

The emergence of a minimal length at the Planck scale is consistent with modern developments in quantum gravity. This is taken into account by transforming the Heisenberg uncertainty principle into the generalized uncertainty principle. Here, the position-momentum commutator is modified accordingly. In this paper, majorization uncertainty relations within the generalized uncertainty principle are considered. Dealing with observables with continuous spectra, each of the axes of interest is divided into a set of non-intersecting bins. Such formulation is consistent with real experiments with a necessarily limited precision. On the other hand, the majorization approach is mainly indicative for high-resolution measurements with sufficiently small bins. Indeed, the effects of the uncertainty principle are brightly manifested just in this case. The current study aims to reveal how the generalized uncertainty principle affects the leading terms of the majorization bound for position and momentum measurements. Interrelations with entropic formulations of this principle are briefly discussed.

New Advances in Quantum Geometry

“…The Issue includes state-of-the-art research contributions in the following areas: the quantum geometry created by quantum matter [1], quantum metric fluctuations [2], cosmology in modified gravity models [3,4], de Sitter gauge theory [5], and matrix theory models of the gravitational interaction [6]. The nature of quantum configurations in phase space [7], momentum operators in intrinsically curved manifolds [8], uncertainty relations in the presence of a minimal length [9], generalized uncertainty black holes [10], and the effects of quantum gravity on mass scales at high energies [11] are also addressed. These fascinating topics, in which geometry, gravity and quantum mechanics are brought together, provide deeper insights into the unsolved mysteries of the gravitational interaction, at the smallest possible scales.…”