Dimensional flow in discrete quantum geometries

Abstract

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 $0<\alpha <d$, we find that the spatial spectral dimension reduces to $d_s\simeq \alpha$ 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 $d_w=2$. Therefore, these quantum geometries may be considered as fractal only when $\alpha =1$, where the "magic number" ${d_s}^{\mathrm{spacetime}}\simeq 2$ for the spectral dimension of space\emph{time}, 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.

11 pages, 6 figures. v2: discussion improved at several points, typos corrected, results and conclusions unchanged

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Cited by in corpus (5)

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• Multifractional theories: an unconventional review
• Multiscale spacetimes from first principles
• Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime
• Deformed symmetries in noncommutative and multifractional spacetimes