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

Abstract

We extend the definition of "spectral dimension" (usually defined for fractal and lattice geometries) to theories on smooth spacetimes with anisotropic scaling. We show that in quantum gravity dominated by a Lifshitz point with dynamical critical exponent z in D+1 spacetime dimensions, the spectral dimension of spacetime is equal to d_s=1+D/z. In the case of gravity in 3+1 dimensions presented in arXiv:0901.3775, which is dominated by z=3 in the UV and flows to z=1 in the IR, the spectral dimension of spacetime flows from d_s=4 at large scales, to d_s=2 at short distances. Remarkably, this is the qualitative behavior of d_s found numerically by Ambjorn, Jurkiewicz and Loll in their causal dynamical triangulations approach to quantum gravity.