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

TL;DR

The paper extends the spectral-dimension observable to smooth spacetimes with anisotropic scaling (Lifshitz gravity) and derives, for a $D$-dimensional spatial geometry, $d_s=1+\frac{D}{z}$. In $3+1$ dimensions with a UV Lifshitz exponent $z=3$, this yields $d_s=2$ in the UV, flowing to $d_s=4$ in the IR as $z\to1$, matching the qualitative behavior seen in causal dynamical triangulations (CDT). The authors argue that CDT's apparent dimensional reduction may reflect a universality class shared with Lifshitz gravity, rather than a dramatic change in topology or discreteness. This work links continuum Lifshitz gravity to lattice CDT results, suggesting that anisotropic scaling provides a natural mechanism for dynamical dimensional reduction in quantum gravity.

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.