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 Horava 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.