Quantum spectral dimension in quantum field theory

TL;DR

This paper redefines the spectral dimension of spacetime within quantum field theory, providing a covariant, probability-based interpretation that clarifies quantum versus classical dimensions and explores their scale-dependent behavior in quantum gravity models.

Contribution

It introduces a new QFT-based interpretation of spectral dimension, resolving issues with negative probabilities and diffusion equations, and applies it to quantum gravity theories showing scale-dependent dimensionality.

Findings

Spectral dimension can deviate from classical topological dimension due to quantum effects.

In higher momentum regimes, the spectral dimension decreases, reaching a universal value of 2 in UV for certain gravity models.

The new interpretation avoids noncovariant diffusion equations and negative probabilities in spectral dimension analysis.

Abstract

We reinterpret the spectral dimension of spacetimes as the scaling of an effective self-energy transition amplitude in quantum field theory (QFT), when the system is probed at a given resolution. This picture has four main advantages: (a) it dispenses with the usual interpretation (unsatisfactory in covariant approaches) where, instead of a transition amplitude, one has a probability density solving a nonrelativistic diffusion equation in an abstract diffusion time; (b) it solves the problem of negative probabilities known for higher-order and nonlocal dispersion relations in classical and quantum gravity; (c) it clarifies the concept of quantum spectral dimension as opposed to the classical one. We then consider a class of logarithmic dispersion relations associated with quantum particles and show that the spectral dimension $d_s$ of spacetime as felt by these quantum probes can deviate from its classical value, equal to the topological dimension $D$. In particular, in the presence of higher momentum powers it changes with the scale, dropping from $D$ in the infrared (IR) to a value $d_s^{\rm UV}\leq D$ in the ultraviolet (UV). We apply this general result to Stelle theory of renormalizable gravity, which attains the universal value $d_s^{\rm UV}=2$ for any dimension $D$.