2013/11/30 by Gianluca Calcagni, Daniele Oriti, Johannes Thürigen · 4 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Mathematical Theories and Applications #Dimension (graph theory) #Lebesgue covering dimension #Loop quantum gravity #Noncommutative and Quantum Gravity Theories #Quantum #Quantum geometry #Quantum gravity #Quantum state #Topology (electrical circuits) #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/0264-9381/31/13/135014
published as Class. Quantum Grav. 31 (2014) 135014 · 39 pages, 18 multiple figures. v2: discussion improved, minor typos corrected
openalex publication_date 2014/06/17 · arxiv created 2014/06/18 · arxiv updated 2014/06/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
The spectral dimension is an indicator of geometry and topology of spacetime and a tool to compare the description of quantum geometry in various approaches to quantum gravity. This is possible because it can be defined not only on smooth geometries but also on discrete (e.g., simplicial) ones. In this paper, we consider the spectral dimension of quantum states of spatial geometry defined on combinatorial complexes endowed with additional algebraic data: the kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the effects of topology and discreteness of classical discrete geometries are studied in a systematic manner. We look for states reproducing the spectral dimension of a classical space in the appropriate regime. We also test the hypothesis that in LQG, as in other approaches, there is a scale dependence of the spectral dimension, which runs from the topological dimension at large scales to a smaller one at short distances. While our results do not give any strong support to this hypothesis, we can however pinpoint when the topological dimension is reproduced by LQG quantum states. Overall, by exploring the interplay of combinatorial, topological and geometrical effects, and by considering various kinds of quantum states such as coherent states and their superpositions, we find that the spectral dimension of discrete quantum geometries is more sensitive to the underlying combinatorial structures than to the details of the additional data associated with them.