Multifractional theories: an unconventional review

TL;DR

This paper provides a comprehensive review and new insights into multifractional spacetime theories, discussing recent developments, theoretical constructions, and implications for quantum gravity, along with experimental bounds.

Contribution

It offers new theoretical results on multifractional derivatives, symmetries, and the relation between different multifractional theories, advancing understanding of quantum gravity implications.

Findings

Impact of flow-equation theorem on dimensional flow

Construction of multifractional derivatives and symmetries

Potential renormalizability of $T_\gamma$ in UV regimes

Abstract

We answer to 72 frequently asked questions about theories of multifractional spacetimes. Apart from reviewing and reorganizing what we already know about such theories, we discuss the physical meaning and consequences of the very recent flow-equation theorem on dimensional flow in quantum gravity, in particular its enormous impact on the multifractional paradigm. We will also get some new theoretical results about the construction of multifractional derivatives and the symmetries in the yet-unexplored theory $T_\gamma$, the resolution of ambiguities in the calculation of the spectral dimension, the relation between the theory $T_q$ with $q$-derivatives and the theory $T_\gamma$ with fractional derivatives, the interpretation of complex dimensions in quantum gravity, the frame choice at the quantum level, the physical interpretation of the propagator in $T_\gamma$ as an infinite superposition of quasiparticle modes, the relation between multifractional theories and quantum gravity, and the issue of renormalization, arguing that power-counting arguments do not capture the exotic properties of extreme UV regimes of multifractional geometry, where $T_\gamma$ may indeed be renormalizable. A careful discussion of experimental bounds and new constraints are also presented.