2012/04/30 by Gianluca Calcagni · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Computer science #Cosmology and Gravitation Theories #Diffusion #Diffusion process #Dimension (graph theory) #Fractal #Geometry #Mathematical analysis #Mathematical and Theoretical Analysis #Mathematics #Multifractal system #Open quantum system #Physics #Pure mathematics #Quantum #Quantum geometry #Quantum gravity #Quantum mechanics #Quantum operation #Quantum spacetime #Scale (ratio) #Spacetime #Statistical physics #Statistics #Stochastic process #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1103/physrevd.86.044021
published as Phys. Rev. D 86, 044021 (2012) · 5 pages, 1 figure. v2: title slightly changed, discussion improved
arxiv created 2012/08/15 · openalex publication_date 2012/08/15 · arxiv updated 2012/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The change of the effective dimension of spacetime with the probed scale is a universal phenomenon shared by independent models of quantum gravity. Using tools of probability theory and multifractal geometry, we show how dimensional flow is controlled by a multiscale fractional diffusion equation, and physically interpreted as a composite stochastic process. The simplest example is a fractional telegraph process, describing quantum spacetimes with a spectral dimension equal to 2 in the ultraviolet and monotonically rising to 4 towards the infrared. The general profile of the spectral dimension of the recently introduced multifractional spaces is constructed for the first time.