# Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime

## TL;DR

This paper demonstrates that measurement uncertainties in quantum gravity can be modeled by a classical, scale-dependent fractal spacetime with fuzzy geometry, linking quantum effects to stochastic spacetime structures.

## Contribution

It shows that quantum measurement uncertainties are reproduced by a classical fractal spacetime with scale-dependent geometry and imaginary dimensions, providing new insights into quantum gravity models.

## Key findings

- Dimensional flow causes intrinsic measurement uncertainty.
- Multi-fractional theories can be constrained by these results.
- Short-scale geometry exhibits stochastic fuzziness and discrete scale invariance.

## Abstract

We show that the uncertainty in distance and time measurements found by the heuristic combination of quantum mechanics and general relativity is reproduced in a purely classical and flat multi-fractal spacetime whose geometry changes with the probed scale (dimensional flow) and has non-zero imaginary dimension, corresponding to a discrete scale invariance at short distances. Thus, dimensional flow can manifest itself as an intrinsic measurement uncertainty and, conversely, measurement-uncertainty estimates are generally valid because they rely on this universal property of quantum geometries. These general results affect multi-fractional theories, a recent proposal related to quantum gravity, in two ways: they can fix two parameters previously left free (in particular, the value of the spacetime dimension at short scales) and point towards a reinterpretation of the ultraviolet structure of geometry as a stochastic foam or fuzziness. This is also confirmed by a correspondence we establish between Nottale scale relativity and the stochastic geometry of multi-fractional models.