Would quantum gravity be renormalized if the number of spacetime dimensions decreased at small distances? In the discussion below (bottom of page on link) if the number of dimensions of spacetime d was lower than 3 at small distances would a theory of quantum gravity be renormalizable? Is is possible that spacetime has fewer than 3 dimensions at the microscopic level?
quantum gravity renormalizarion
 A: The idea that spacetime dimensionality becomes lower at small distances/high energies is somewhat vague, but  there are hints that this is indeed what happens in several different approaches to quantum gravity.
A non-technical review of this idea could be found in the paper:

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*Carlip, S. (2017). Dimension and dimensional reduction in quantum gravity. Classical and Quantum Gravity, 34(19), 193001, doi:10.1088/1361-6382/aa8535, arXiv:1705.05417.

Abstract

A number of very different approaches to quantum gravity contain a common thread, a hint that spacetime at very short distances becomes effectively two dimensional. I review this evidence, starting with a discussion of the physical meaning of ‘dimension’ and concluding with some speculative ideas of what dimensional reduction might mean for physics.

From the paper:

We have seen that many approaches to quantum gravity show indications of dimensional reduction
near the Planck scale. Taken individually, none of these hints is terribly convincing. Perhaps the
best evidence comes from asymptotic safety, in which the argument for two-dimensional behavior
at the ultraviolet fixed point is compelling, and causal dynamical triangulations, in which the
evidence for flow of the spectral dimension is extremely strong. But for this evidence to be truly
persuasive, we would have to actually know that quantum gravity has an interacting ultraviolet
fixed point, as required by asymptotic safety, or that causal dynamical triangulations has the right
continuum limit; that is, we would have to know how to quantize gravity.
Taken as a body, though, these hints become quite a bit more compelling. It seems rather
unlikely that so many different approaches to quantum gravity would converge on the same result
merely by accident. If this convergence is more than coincidence, though, it ought to be possible
to to find a common thread, a single origin for dimensional reduction that is shared by all of the
various approaches.

