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.

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

References in corpus (16)

• Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point
• A Gravity Theory on Noncommutative Spaces
• Fractal properties of quantum spacetime
• Kappa-Minkowski space-time and the star product realizations
• Dimension and Dimensional Reduction in Quantum Gravity
• Field theory on $\kappa$--Minkowski space revisited: Noether charges and breaking of Lorentz symmetry
• From noncommutative kappa-Minkowski to Minkowski space-time
• Thermodynamics of photons on fractals
• Multifractional theories: an unconventional review
• Diffusion on $\kappa$-Minkowski space
• Dimensional flow in discrete quantum geometries
• Diffusion in quantum geometry
• Multiscale spacetimes from first principles
• Evidence for Asymptotic Safety from Dimensional Reduction in Causal Dynamical Triangulations
• Spectral Action in Noncommutative Geometry
• UV dimensional reduction to two from group valued momenta

Cited by in corpus (1)

• Ultraviolet dimensional reduction of spacetime with zero-point length