Spectral dimensions and dimension spectra of quantum spacetimes

Abstract

Different approaches to quantum gravity generally predict that the dimension of spacetime at the fundamental level is not 4. The principal tool to measure how the dimension changes between the IR and UV scales of the theory is the spectral dimension. On the other hand, the noncommutative-geometric perspective suggests that quantum spacetimes ought to be characterised by a discrete complex set -- the dimension spectrum. Here we show that these two notions complement each other and the dimension spectrum is very useful in unravelling the UV behaviour of the spectral dimension. We perform an extended analysis highlighting the trouble spots and illustrate the general results with two concrete examples: the quantum sphere and the $\kappa$-Minkowski spacetime, for a few different Laplacians. In particular, we find out that the spectral dimensions of the former exhibit log-periodic oscillations, the amplitude of which decays rapidly as the deformation parameter tends to the classical value. In contrast, no such oscillations occur for either of the three considered Laplacians on the $\kappa$-Minkowski spacetime.

arXiv comment35 pages, 7 figures, v2 some comments and references added, summary extended, title changed

References in corpus (60)

• Noncommutative Perturbative Dynamics
• Heat kernel expansion: user's manual
• Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Quantum Spacetime Phenomenology
• Super-renormalizable Quantum Gravity
• Discrete scale invariance and complex dimensions
• Spectral Dimension of the Universe
• A Gravity Theory on Noncommutative Spaces
• Fractal Spacetime Structure in Asymptotically Safe Gravity
• Quantum $\kappa$-Poincare in Any Dimensions
• Fractal properties of quantum spacetime
• De Sitter space as an arena for Doubly Special Relativity
• Kappa-Minkowski space-time and the star product realizations
• Doubly Special Relativity and de Sitter space
• Spectral geometry as a probe of quantum spacetime
• Dimension and Dimensional Reduction in Quantum Gravity
• Noncommutative Differential Calculus on the Kappa-Minkowski Space
• Field theory on $\kappa$--Minkowski space revisited: Noether charges and breaking of Lorentz symmetry
• Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification
• From noncommutative kappa-Minkowski to Minkowski space-time
• Inner Fluctuations in Noncommutative Geometry without the first order condition
• Heat Kernel Approach in Quantum Field Theory
• Thermodynamics of photons on fractals
• Multifractional theories: an unconventional review
• Diffusion on $\kappa$-Minkowski space
• Gravitational effective action and entanglement entropy in UV modified theories with and without Lorentz symmetry
• Heat Kernels and Zeta Functions on Fractals
• Dimensional flow in discrete quantum geometries
• Anomalous dimension in semiclassical gravity
• Multiscale spacetimes from first principles
• Diffusion in quantum geometry
• Evidence for Asymptotic Safety from Dimensional Reduction in Causal Dynamical Triangulations
• Dimensional reduction in causal set gravity
• Spectral triples for the Sierpinski Gasket
• Complex dimensions and their observability
• Star product realizations of kappa-Minkowski space
• Wick Rotation and Fermion Doubling in Noncommutative Geometry
• Spectral Dimension from Nonlocal Dynamics on Causal Sets
• Towards the map of quantum gravity
• Emergence of Spacetime in a restricted Spin-foam model
• Noncommutative fields and the short-scale structure of spacetime
• Imprint of quantum gravity in the dimension and fabric of spacetime
• A modular spectral triple for $\kappa$-Minkowski space
• Differential calculi on quantum Minkowski space
• Spectral geometry of the Moyal plane with harmonic propagation
• Spectral estimators for finite non-commutative geometries
• Local Index Formula on the Equatorial Podles Sphere
• $\kappa$-Minkowski star product in any dimension from symplectic realization
• Geometry of Quantum Spheres
• Asymptotic and exact expansions of heat traces
• Heat trace and spectral action on the standard Podles sphere
• Spectral Action in Noncommutative Geometry
• Wodzicki residue and minimal operators on a noncommutative 4-dimensional torus
• Quantum $D=3$ Euclidean and Poincaré symmetries from contraction limits
• Discrete scale invariance in holography revisited
• Ultralocal nature of geometrogenesis
• Finite entanglement entropy and spectral dimension in quantum gravity
• UV dimensional reduction to two from group valued momenta
• The Moyal Sphere

Cited by in corpus (9)

• Emergence of the Circle in a Statistical Model of Random Cubic Graphs
• Decoherence and thermalization of Unruh-DeWitt detector in arbitrary dimensions
• Generalised spectral dimensions in non-perturbative quantum gravity
• Ultraviolet dimensional reduction of spacetime with zero-point length
• Curvature effects in the spectral dimension of spin foams
• Quantum gravity and gravitational-wave astronomy
• Quantum Field Theory on Multifractal Spacetime: Varying Dimension and Ultraviolet Completeness
• Newtonian approximation in (1 + 1) dimensions
• Minimal Proper-time in Quantum Field Theory