Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point

We extend the definition of “spectral dimension” $d_s$ (usually defined for fractal and lattice geometries) to theories in spacetimes with anisotropic scaling. We show that in gravity with dynamical critical exponent $z$ in $D+1$ dimensions, the spectral dimension of spacetime is $d_s=1+\frac{D}{z}$. In the case of gravity in $3+1$ dimensions with $z=3$ in the UV which flows to $z=1$ in the IR, the spectral dimension changes from $d_s=4$ at large scales to $d_s=2$ at short distances. Remarkably, this is the behavior found numerically by Ambjørn et al. in their causal dynamical triangulations approach to quantum gravity.

Figure 1

(a) The preferred foliation by time slices in the continuum approach of gravity with anisotropic scaling, and (b) a characteristic lattice configuration in the CDT approach.