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.

Highlights

  • After many years of research, we are not yet close to an acknowledged unique quantum theory of gravity, partly because of the lack of experimental guidance

  • 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 and has non-zero imaginary dimension, corresponding to a discrete scale invariance at short distances

  • Dimensional flow can manifest itself as an intrinsic measurement uncertainty and, measurement-uncertainty estimates are generally valid because they rely on this universal property of quantum geometries

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Summary

Introduction

After many years of research, we are not yet close to an acknowledged unique quantum theory of gravity, partly because of the lack of experimental guidance. For special values of α in Eq (1), this multi-fractional contribution can be reinterpreted as an intrinsic uncertainty (or fuzziness, in QG jargon) on the measurement of spacetime distances exactly of the same type encountered in a standard (i.e., non-multi-scale) model where both QM and GR are taken into account [33, 34]. This suggests that classical multifractional models in Minkowski spacetime (i.e., in the absence of curvature) partially encode both QM and GR effects, and that they do so thanks to dimensional flow. A condensed presentation can be found in [44]

Review of the estimates
Deterministic view
Stochastic view
Stochastic view with multi-fractional derivatives
Stochastic view with random measure
Summary
Related proposals
Avoiding observational constraints on multi-fractional theories
Structure of multi-scale spacetimes
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