Fractional and noncommutative spacetimes

Abstract

We establish a mapping between fractional and noncommutative spacetimes in configuration space. Depending on the scale at which the relation is considered, there arise two possibilities. For a fractional spacetime with log-oscillatory measure, the effective measure near the fundamental scale determining the log-period coincides with the non-rotation-invariant but cyclicity-preserving measure of κ-Minkowski. At scales larger than the log-period, the fractional measure is averaged and becomes a power-law with real exponent. This can be also regarded as the cyclicity-inducing measure in a noncommutative spacetime defined by a certain nonlinear algebra of the coordinates, which interpolates between κ-Minkowski and canonical spacetime. These results are based upon a braiding formula valid for any nonlinear algebra which can be mapped onto the Heisenberg algebra.

Standard Model in multiscale theories and observational constraints

“…In this context, we focus on theories of multiscale spacetimes [4,[30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. These have been proposed either as stand-alone models of exotic geometry [31,32,40,43] or as an effective means to study, in a controlled manner, the change of dimensionality with the probed scale (known as dimensional flow 1 of these models and of their status).…”

“…This modulation factor includes logarithmic oscillations and a fundamental scale l ∞ much smaller than l à , possibly of order of the Planck scale [33]. We will not use log-oscillating measures in the bulk of this paper, as the multifractional binomial measure (10) will suffice for our purpose.…”

Diffusion in multiscale spacetimes

“…In general, one of the indicators characterizing quantum geometry, the spectral dimension d S of spacetime, changes with the scale, running from d S 2 (or exactly d S = 2) in the ultraviolet (UV) to the usual, classical value d S ∼ 4 in the infrared (IR). Numerical and analytic examples can be found in causal dynamical triangulations (CDT) [4,5], random combs [6,7] and random multigraphs [8,9] (both sharing some properties with CDT), quantum Einstein gravity (QEG, also called asymptotic safety) [10,11], spin foams [12][13][14][15], Hořava-Lifshitz gravity [16,17], noncommutative geometry at the fundamental [18,19] and effective [20][21][22] levels, field theory on multifractal spacetimes [23][24][25] (in particular, in the realization within multifractional geometry [26][27][28][29][30][31]), and nonlocal super-renormalizable quantum gravity [32][33][34].…”

“…Therefore, it can be regarded either as an independent proposal for a fundamental theory or an effective framework wherein to better understand the multiscale geometry of the other approaches (examples are [21,35,36]). For this reason, we believe it is important to exploit the tools available in multifractional spacetimes as much as possible.…”

Asymptotic Safety, Fractals, and Cosmology

“…For similar work on fractal features in different approaches we must refer to the literature [69][70][71][72][73][74][75][76][77][78][79][80][81][82]. (3) As for possible physics implications of the RG flow predicted by QEG, ideas from particle physics, in particular the "RG improvement", have been employed in order to study the leading quantum gravity effects in black holes [83,84], cosmological space-times [51,52,[85][86][87][88][89][90][91][92][93] or possible observable signatures from Asymptotic Safety at the LHC [94][95][96][97].…”