From dispersion relations to spectral dimension - and back again

Abstract

The so-called spectral dimension is a scale-dependent number associated with both geometries and field theories that has recently attracted much attention, driven largely though not exclusively by investigations of causal dynamical triangulations (CDT) and Horava gravity as possible candidates for quantum gravity. We advocate the use of the spectral dimension as a probe for the kinematics of these (and other) systems in the region where spacetime curvature is small, and the manifold is flat to a good approximation. In particular, we show how to assign a spectral dimension (as a function of so-called diffusion time) to any arbitrarily specified dispersion relation. We also analyze the fundamental properties of spectral dimension using extensions of the usual Seeley-DeWitt and Feynman expansions, and by saddle point techniques. The spectral dimension turns out to be a useful, robust and powerful probe, not only of geometry, but also of kinematics.

V1: 26 pages. V2: 28 pages. Several references added. Some references updated. Typos fixed. Minor changes in presentation. This version closely follows the version published in Physical Review D

References in corpus (16)

• Quantum Gravity at a Lifshitz Point
• Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point
• Phenomenologically viable Lorentz-violating quantum gravity
• Lorentz symmetry breaking as a quantum field theory regulator
• Horava-Lifshitz gravity: a status report
• Fractal properties of quantum spacetime
• Hawking-like radiation from evolving black holes and compact horizonless objects
• Evidence for Asymptotic Safety from Lattice Quantum Gravity
• Spectral dimension as a probe of the ultraviolet continuum regime of causal dynamical triangulations
• Projectable Horava-Lifshitz gravity in a nutshell
• Causal Dynamical Triangulations and the Quest for Quantum Gravity
• Lower-dimensional Horava-Lifshitz gravity
• (2+1)-Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations
• CDT---an Entropic Theory of Quantum Gravity
• On spectral geometry approach to Horava-Lifshitz gravity: Spectral dimension
• Continuum Random Combs and Scale Dependent Spectral Dimension

Cited by in corpus (49)

• Nonperturbative Quantum Gravity
• Hořava Gravity at a Lifshitz Point: A Progress Report
• Dimension and Dimensional Reduction in Quantum Gravity
• Spectral dimension as a probe of the ultraviolet continuum regime of causal dynamical triangulations
• Multifractional theories: an unconventional review
• Rainbow gravity and scale-invariant fluctuations
• Diffusion on $\kappa$-Minkowski space
• Diffusion in multiscale spacetimes
• Dimensional reduction in the sky
• Diffusion in quantum geometry
• Intermediate inflation from rainbow gravity
• Planck-scale dimensional reduction without a preferred frame
• Quantum Gravity signatures in the Unruh effect
• Laplacians on discrete and quantum geometries
• Spectral dimension and diffusion in multi-scale spacetimes
• Noncommutative fields and the short-scale structure of spacetime
• Spectral dimensions from the spectral action
• Metric dimensional reduction at singularities with implications to Quantum Gravity
• Dynamical dimensional reduction in toy models of 4D causal quantum gravity
• Thermal dimension of quantum spacetime
• Gravity as the breakdown of conformal invariance
• Quantum spectral dimension in quantum field theory
• Spectral estimators for finite non-commutative geometries
• Observational constraints on dual intermediate inflation
• Signal propagation on $\kappa$-Minkowski spacetime and non-local two-point functions
• On the UV dimensions of Loop Quantum Gravity
• Quantum gravity without vacuum dispersion
• Hypothesis on the Nature of Time
• Spin effects and compactification
• Causal hierarchy in modified gravity
• Extrinsic curvature in 2-dimensional Causal Dynamical Triangulation
• Singular General Relativity
• Dimensional flow in the kappa-deformed space-time
• Regularization versus renormalization: Why are Casimir energy differences so often finite?
• Primordial perturbations in a rainbow universe with running Newton constant
• Fractional quantum mechanics meets quantum gravity phenomenology
• Modified dispersion relations, inflation and scale invariance
• Some remarks on Relativistic Diffusion and the Spectral Dimension Criterion
• Geometry of fractional spaces
• Finite entanglement entropy and spectral dimension in quantum gravity
• UV dimensional reduction to two from group valued momenta
• From Causal Dynamical Triangulations To Astronomical Observations
• Localization and diffusion in polymer quantum field theory
• The (2+1)-dim Axial Universes -- Solutions to the Einstein Equations, Dimensional Reduction Points, and Klein-Fock-Gordon Waves
• Spectral dimension in graph models of causal quantum gravity
• Free Energy on a Cycle Graph and Trigonometric Deformation of Heat Kernel Traces on Odd Spheres
• Aspects of dynamical dimensional reduction in multigraph ensembles of CDT
• A Falsifiable Alternative to General Relativity
• Three-dimensional gravity and deformations of relativistic symmetries