Fractal Space-Time from Spin-Foams

In this paper we perform the calculation of the spectral dimension of spacetime in 4d quantum gravity using the Barrett-Crane (BC) spinfoam model. We realize this considering a very simple decomposition of the 4d spacetime already used in the graviton propagator calculation and we introduce a boundary state which selects a classical geometry on the boundary. We obtain that the spectral dimension of the spacetime runs from $\approx 2$ to 4, across a $\approx 1.5$ phase, when the energy of a probe scalar field decreases from high $E\lesssim E_P\mathrm{/}25$ to low energy. The spectral dimension at the Planck scale $E\approx E_P$ depends on the areas spectrum used in the calculation. For three different spectra $l_P^2\sqrt{j(j+1)}$, $l_P^2(2j+1)$ and $l_P^2 j$wefindrespectivelydimension$\approx 2.31$, 2.45 and 2.08.