Would quantum gravity be renormalized if the number of spacetime dimensions decreased at small distances?

Question

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

Answer 1

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:

• 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.