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.

arXiv comment1+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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