2014/12/31 by Gianluca Calcagni, Daniele Oriti, Johannes Thürigen · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Dimension (graph theory) #Effective dimension #Flow (mathematics) #Fractal #Fractal dimension #Geometry #Hausdorff dimension #Loop quantum gravity #Mathematical analysis #Mathematics #Minkowski–Bouligand dimension #Noncommutative and Quantum Gravity Theories #Physics #Pure mathematics #Quantum #Quantum gravity #Quantum mechanics #Superposition principle #Theoretical physics #hep-th #math-ph #math.MP
paper · pdf · doi:10.1103/physrevd.91.084047
published as Phys. Rev. D 91, 084047 (2015) · 11 pages, 6 figures. v2: discussion improved at several points, typos corrected, results and conclusions unchanged
openalex publication_date 2015/04/20 · arxiv created 2015/04/21 · arxiv updated 2015/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In various theories of quantum gravity, one observes a change in the spectral dimension from the topological spatial dimension d at large length scales to some smaller value at small, Planckian scales. While the origin of such a flow is well understood in continuum approaches, in theories built on discrete structures a firm control of the underlying mechanism is still missing. We shed some light on the issue by presenting a particular class of quantum geometries with a flow in the spectral dimension, given by superpositions of states defined on regular complexes. For particular superposition coefficients parametrized by a real number 0<\ensuremathα<d, we find that the spatial spectral dimension reduces to dS\ensuremath≃\ensuremathα at small scales. The spatial Hausdorff dimension of such class of states varies between 1 and d, while the walk dimension takes the usual value dW=2. Therefore, these quantum geometries may be considered as fractal only when \ensuremathα=1, where the ``magic number'' DS\ensuremath≃2 for the spectral dimension of spacetime, appearing so often in quantum gravity, is reproduced as well. These results apply, in particular, to special superpositions of spin-network states in loop quantum gravity, and they provide more solid indications of dimensional flow in this approach.