A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle

Abstract

In this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator, $[\hat{X},\hat{P}]=i\hslash (1+\beta p^2)$, may have different physical consequences such as having no minimal length at all. These differences depend on how the position and/or momentum operators are modified rather than only on the resulting modified commutator. This provides guidance when constructing GUP models since it distinguishes those GUPs that have a minimal length scale, as suggested by some broad arguments about quantum gravity, versus GUPs without a minimal length scale.

arXiv comment14 pages, 3 figures. Published in Universe

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• Influence of GUP corrected Casimir energy on zero tidal force wormholes in modified teleparallel gravity with matter coupling
• Fermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP case
• Unveiling Phase Space Modifications: A Clash of Modified Gravity and the Generalized Uncertainty Principle
• Constraining Snyder and GUP models with low-mass stars
• Generalized Extended Uncertainty Principles, Liouville theorem and density of states: Snyder-de Sitter and Yang models
• The more things change the more they stay the same: Minimum lengths with unmodified uncertainty principle and dispersion relation
• On Some Quantum Correction to the Coulomb Potential in Generalized Uncertainty Principle Approach
• Testing quantum gravity with dilute dipolar Bose gases
• Bose-Einstein Condensate and Liquid Helium He${}^4$: Implications of GUP and Modified Gravity Correspondence
• Lorentz-covariant sampling theory for fields