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

Abstract

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 Ambj\o{}rn et al. in their causal dynamical triangulations approach to quantum gravity.