Multifractional theories: an unconventional review

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.

1+106 pages, 3 figures, 9 tables, 245 references. Review article (with several important novelties) through 72 questions; in some of them, there is text overlap with papers by the author, all indicated in the text. v2: references added, minor typos corrected, answers to questions 01, 59 and 68 expanded. v3: minor typos corrected

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Cited by in corpus (20)

• Dimension and Dimensional Reduction in Quantum Gravity
• Prospecting Black Hole Thermodynamics with Fractional Quantum Mechanics
• Cosmology under the fractional calculus approach
• Multiscale spacetimes from first principles
• Revisiting Fractional Cosmology
• First Combined Study on LIV from Observations of Energy-dependent Time Delays from Multiple-type Gamma-ray Sources -- Part I. Motivation, Method Description and Validation through Simulations of H.E.S.S., MAGIC and VERITAS Datasets
• Exact solutions and cosmological constraints in fractional cosmology
• Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime
• Newtonian Fractional-Dimension Gravity and Rotationally Supported Galaxies
• Emergence of fractal cosmic space from fractional quantum gravity
• Modified Friedmann equations from fractional entropy
• Spectral dimensions and dimension spectra of quantum spacetimes
• Representations of the fractional d'Alembertian and initial conditions in fractional dynamics
• Fractional Einstein field equations in $2+1$ dimensional spacetime
• Entanglement entropy, scale-dependent dimensions and the origin of gravity
• Gravitational Foundations and Exact Solutions in $n$-Dimensional Fractional Cosmology
• Quantum Field Theory on Multifractal Spacetime: Varying Dimension and Ultraviolet Completeness
• Hawking area law in quantum gravity
• Cosmology of fractional gravity
• Relativistic Fractional-Dimension Gravity