Fractal Properties of Quantum Spacetime

We show that, in general, a spacetime having a quantum group symmetry has also a scale-dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what is observed in some approaches to quantum gravity. In particular, we analyze the cases of a quantum sphere and of $\kappa$-Minkowski spacetime, the latter being relevant in the context of quantum gravity.

Figure 1

(a) The spectral dimension of a unit sphere $SU(2)/U(1)$. (b) The case of a quantum sphere $S{U}_q(2)/U(1)$, for $z=0.01$.

Figure 2

A plot of the spectral dimension $d_s$ of $\kappa$-Minkowski space for $\kappa =1$ as a function of the diffusion time $s$. For comparison, we plot also the constant behavior of the spectral dimension of classical Minkowski space ($d_s=4$).