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 $0<\ensuremath{\alpha}<d$, we find that the spatial spectral dimension reduces to ${d}_{\mathrm{S}}\ensuremath{\simeq}\ensuremath{\alpha}$ 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 ${d}_{\mathrm{W}}=2$. Therefore, these quantum geometries may be considered as fractal only when $\ensuremath{\alpha}=1$, where the ``magic number'' ${D}_{\mathrm{S}}\ensuremath{\simeq}2$ 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.