Introduction to multifractional spacetimes

TL;DR

This paper reviews the construction and properties of multifractional spacetimes with scale-dependent dimensions, discrete ultraviolet symmetries, and potential perturbative renormalizability, connecting to quantum gravity theories.

Contribution

It provides an informal overview of multifractional geometries, highlighting their scale-dependent features and potential for renormalizable field theories including gravity.

Findings

Multifractional spacetimes have scale-dependent dimensions.

They exhibit discrete symmetries in the ultraviolet.

Field theories on these geometries may be perturbatively renormalizable.

Abstract

We informally review the construction of spacetime geometries with multifractal and, more generally, multiscale properties. Based on fractional calculus, these continuous spacetimes have their dimension changing with the scale; they display discrete symmetries in the ultraviolet and ordinary Poincar\'e symmetries in the infrared. Under certain reasonable assumptions, field theories (including gravity) on multifractional geometries are generally argued to be perturbatively renormalizable. We also sketch the relation with other field theories of quantum gravity based on the renormalization group.