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A hallmark of non-perturbative theories of quantum gravity is the absence of a fixed background geometry, and therefore the absence in a Planckian regime of any notion of length or scale that is defined a priori. This has potentially far-reaching consequences for the application of renormalization group methods \`a la Wilson, which rely on these notions in a crucial way. We review the status quo of attempts in the Causal Dynamical Triangulations (CDT) approach to quantum gravity to find an ultraviolet fixed point associated with the second-order phase transitions observed in the lattice theory. Measurements of the only invariant correlator currently accessible, that of the total spatial three-volume, has not produced any evidence of such a fixed point. A possible explanation for this result is our incomplete and perhaps na\"ive understanding of what constitutes an appropriate notion of (quantum) length near the Planck scale.
Ambjørn et al. (Thu,) studied this question.
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