Fractal properties of particle paths due to generalised uncertainty relations

Abstract

We determine the Hausdorff dimension of a particle path, $D_{\mathrm{H}}$, in the recently proposed `smeared space' model of quantum geometry. The model introduces additional degrees of freedom to describe the quantum state of the background and gives rise to both the generalised uncertainty principle (GUP) and extended uncertainty principle (EUP) without introducing modified commutation relations. We compare our results to previous studies of the Hausdorff dimension in GUP models based on modified commutators and show that the minimum length enters the relevant formulae in a different way. We then determine the Hausdorff dimension of the particle path in smeared momentum space, ${\tilde{D}}_{\mathrm{H}}$, and show that the minimum momentum is dual to the minimum length. For sufficiently coarse grained paths, $D_{\mathrm{H}}={\tilde{D}}_{\mathrm{H}}=2$, as in canonical quantum mechanics. However, as the resolutions approach the minimum scales, the dimensions of the paths in each representation differ, in contrast to their counterparts in the canonical theory. The GUP-induced corrections increase $D_{\mathrm{H}}$ whereas the EUP-induced corrections decrease ${\tilde{D}}_{\mathrm{H}}$, relative to their canonical values, and the extremal case corresponds to $D_{\mathrm{H}}=3$, ${\tilde{D}}_{\mathrm{H}}=1$. These results show that the GUP and the EUP affect the fractal properties of the particle path in fundamentally different, yet complimentary, ways.

arXiv comment19 pages, 2 figures, 2 tables. Published version

References in corpus (37)

• In the Realm of the Hubble tension $-$ a Review of Solutions
• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• Quantum gravity and minimum length
• Tensions between the Early and the Late Universe
• Generalized Uncertainty Principle in Quantum Gravity from Micro-Black Hole Gedanken Experiment
• Minimal Length Scale Scenarios for Quantum Gravity
• The Area of a Rough Black Hole
• On Gravity and the Uncertainty Principle
• Quantum mechanics and the covariance of physical laws in quantum reference frames
• Generalized Uncertainty Principle: Approaches and Applications
• 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
• Review on Generalized Uncertainty Principle
• Barrow entropic dark energy: A member of generalized holographic dark energy family
• Natural extension of the Generalised Uncertainty Principle
• Uncertainty Relation on World Crystal and its Applications to Micro Black Holes
• Generalized Uncertainty Principle and Corpuscular Gravity
• Generalized Uncertainty Principle, Black Holes, and White Dwarfs: A Tale of Two Infinities
• Generalised uncertainty relations from superpositions of geometries
• Barrow Entropy Cosmology: an observational approach with a hint of stability analysis
• Baryon asymmetry from Barrow entropy: theoretical predictions and observational constraints
• The deformation parameter of the generalized uncertainty principle
• Hints of Modified Gravity in Cosmos and in the Lab?
• Sub-millimeter Spatial Oscillations of Newton's Constant: Theoretical Models and Laboratory Tests
• Generalized uncertainty principle and Asymptotic Safe gravity
• Generalized uncertainty principle with maximal observable momentum and no minimal length indeterminacy
• Mixing quantum and classical mechanics and uniqueness of Planck's constant
• Generalised uncertainty relations for angular momentum and spin in quantum geometry
• Constraints on Spatially Oscillating Sub-mm Forces from the Stanford Levitated Microsphere Experiment Data
• Hausdorff dimension of a particle path in a quantum manifold
• The minimum mass of a spherically symmetric object in $D$-dimensions, and its implications for the mass hierarchy problem
• Reconciling a quantum gravity minimal length with lack of photon dispersion
• Does Compton/Schwarzschild duality in higher dimensions exclude TeV quantum gravity?
• How does the Planck scale affect qubits?
• Is there a connection between "dark" and "light" physics?
• A solution to the soccer ball problem for generalised uncertainty relations
• Why space could be quantised on a different scale to matter

Cited by in corpus (2)

• Problems with Modified Commutators
• Measurement-Induced Perturbations of Hausdorff Dimension in Quantum Paths