Simple arguments based on the general properties of quantum fluctuations have
been recently shown to imply that quantum fluctuations of spacetime obey the
same scaling laws of the velocity fluctuations in a homogeneous incompressible
turbulent flow, as described by Kolmogorov 1941 (K41) scaling theory. Less
noted, however, is the fact that this analogy rules out the possibility of a
fractal quantum spacetime, in contradiction with growing evidence in quantum
gravity research. In this Note, we show that the notion of a fractal quantum
spacetime can be restored by extending the analogy between turbulence and
quantum gravity beyond the realm of K41 theory. In particular, it is shown that
compatibility of a fractal quantum-space time with the recent Horava-Lifshitz
scenario for quantum gravity, implies singular quantum wavefunctions. Finally,
we propose an operational procedure, based on Extended Self-Similarity
techniques, to inspect the (multi)-scaling properties of quantum gravitational
fluctuations.Comment: Sliglty modified version of the article about to appear in IJMP