Discrete to continuum transition in multifractal spacetimes

TL;DR

This paper proposes a field theory on multifractal spacetimes with a variable Hausdorff dimension, revealing a transition from discrete microscopic structures to an effective continuum, and exploring implications for quantum gravity and noncommutative theories.

Contribution

It introduces a multifractal spacetime model with a varying measure, capturing the transition from discrete to continuum regimes and connecting to quantum gravity features.

Findings

Demonstrates a scale-dependent change in effective spacetime dimensionality.

Shows a transition from discrete symmetries to continuum behavior.

Explores implications for noncommutative field theories.

Abstract

We outline a field theory on a multifractal spacetime. The measure in the action is characterized by a varying Hausdorff dimension and logarithmic oscillations governed by a fundamental physical length. A fine hierarchy of length scales identifies different regimes, from a microscopic structure with discrete symmetries to an effectively continuum spacetime. Thanks to general arguments from fractal geometry, this scenario explicitly realizes two indirect or conjectured features of most quantum gravity models: a change of effective spacetime dimensionality with the probed scale, and the transition from a fundamentally discrete quantum spacetime to the continuum. It also allows us to probe ultramicroscopic scales where spectral methods based on ordinary geometry typically fail. Consequences for noncommutative field theories are discussed.