From dispersion relations to spectral dimension - and back again

TL;DR

This paper explores the spectral dimension as a tool for probing the kinematics of spacetime and field theories, establishing methods to assign and analyze it based on dispersion relations and fundamental mathematical expansions.

Contribution

It introduces a way to assign spectral dimension to any dispersion relation and analyzes its properties using advanced mathematical techniques, enhancing its utility as a probe.

Findings

Spectral dimension can be assigned to arbitrary dispersion relations.

Spectral dimension is a robust probe of both geometry and kinematics.

Mathematical analysis confirms the fundamental properties of spectral dimension.

Abstract

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 (CDT) 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 saddle point techniques. The spectral dimension turns out to be a useful, robust and powerful probe, not only of geometry, but also of kinematics.