From dispersion relations to spectral dimension—and back again

The so-called spectral dimension is a scale-dependent number associated with both geometries and field theories that has recently attracted much attention, driven largely, though not exclusively, by investigations of causal dynamical triangulations and Ho {r} {r}{}ava gravity as possible candidates for quantum gravity. We advocate the use of the spectral dimension as a probe for the kinematics of these (and other) systems in the region where spacetime curvature is small, and the manifold is flat to a good approximation. In particular, we show how to assign a spectral dimension (as a function of so-called diffusion time) to any arbitrarily specified dispersion relation. We also analyze the fundamental properties of spectral dimension using extensions of the usual Seeley--DeWitt and Feynman expansions and by using saddle point techniques. The spectral dimension turns out to be a useful, robust, and powerful probe, not only of geometry, but also of kinematics.