Hausdorff dimension of a particle path in a quantum manifold
arXiv:1009.3267 · doi:10.1103/PhysRevD.83.024017
Abstract
After recalling the concept of the Hausdorff dimension, we study the fractal properties of a quantum particle path. As a novelty we consider the possibility for the space where the particle propagates to be endowed with a quantum-gravity-induced minimal length. We show that the Hausdorff dimension accounts for both the quantum mechanics uncertainty and manifold fluctuations. In addition the presence of a minimal length breaks the self-similarity property of the erratic path of the quantum particle. Finally we establish a universal property of the Hausdorff dimension as well as the spectral dimension: They both depend on the amount of resolution loss which affects both the path and the manifold when quantum gravity fluctuations occur.
7 pages, 4 figure, updated version which matches that published on Physical Review D
References in corpus (31)
- String Theory and Noncommutative Geometry
- Hilbert Space Representation of the Minimal Length Uncertainty Relation
- Noncommutative geometry inspired Schwarzschild black hole
- Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review
- Spectral Dimension of the Universe
- Noncommutative geometry inspired charged black holes
- Fractal Spacetime Structure in Asymptotically Safe Gravity
- Noncommutative Black Hole Thermodynamics
- Non-commutative geometry inspired higher-dimensional charged, black holes
- Charged rotating noncommutative black holes
- Noncommutative geometry inspired dirty black holes
- Fractal properties of quantum spacetime
- Fractal Structure of Loop Quantum Gravity
- A model of radiating black hole in noncommutative geometry
- Noncommutative Inspired Black Holes in Extra Dimensions
- "Kerrr" black hole: the Lord of the String
- Spectral dimension of a quantum universe
- Entropic force, noncommutative gravity and ungravity
- Trace Anomaly in Quantum Spacetime Manifold
- The decay-time of non-commutative micro-black holes
- Noncommutative geometry inspired black holes in higher dimensions at the LHC
- A minimal length versus the Unruh effect
- Large extra dimensions and small black holes at the LHC
- Fuzziness at the horizon
- Hausdorff dimension of a quantum string
- Un-spectral dimension and quantum spacetime phases
- Fuzzy dimensions and Planck's Uncertainty Principle for p-branes
- Minimal Scales from an Extended Hilbert Space
- Particle production and transplanckian problem on the non-commutative plane
- Loop Quantum Mechanics and the Fractal Structure of Quantum Spacetime
- How to Measure the Fractal Geometry of the Relativistic Fermion Propagator
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- Geometry and field theory in multi-fractional spacetime
- Physics on Smallest Scales - An Introduction to Minimal Length Phenomenology
- Hawking emission from quantum gravity black holes
- Neutrino oscillations as a novel probe for a minimal length
- Aspects of noncommutative (1+1)-dimensional black holes
- Regular black holes in UV self-complete quantum gravity
- Quantum mechanics in fractional and other anomalous spacetimes
- Black-hole entropy and minimal diffusion
- Spectral dimension and diffusion in multi-scale spacetimes
- Self-similarity, conservation of entropy/bits and the black hole information puzzle
- Hypothesis on the Nature of Time
- Why space could be quantised on a different scale to matter
- Ultraviolet dimensional reduction of spacetime with zero-point length
- A New Approach to Generalised Uncertainty Relations
- Non-Commutative space-time and Hausdorff dimension
- Metamaterial model of fractal time
- Problems with Modified Commutators
- Fractal properties of particle paths due to generalised uncertainty relations
- Quantum Field Theory on Multifractal Spacetime: Varying Dimension and Ultraviolet Completeness
- Minimal Proper-time in Quantum Field Theory
- Minimal Proper Time and Deterministic Microstates: Emergent Quantum Fields and Relativistic Spacetime
- Measurement-Induced Perturbations of Hausdorff Dimension in Quantum Paths