Self-completeness and spontaneous dimensional reduction

TL;DR

This paper explores the relationship between spontaneous dimensional reduction and self-completeness in quantum gravity, showing that in higher dimensions gravity remains self-complete, while in lower dimensions it does not, with implications for Planck-scale physics.

Contribution

It establishes an XOR relationship between self-completeness and dimensional reduction, clarifying their roles in quantum gravity scenarios across different dimensions.

Findings

Gravity in >4D remains self-complete.

In <4D, gravity is not self-complete.

Dimensional reduction and self-completeness are mutually exclusive.

Abstract

A viable quantum theory of gravity is one of the biggest challenges facing physicists. We discuss the confluence of two highly expected features which might be instrumental in the quest of a finite and renormalizable quantum gravity -- spontaneous dimensional reduction and self-completeness. The former suggests the spacetime background at the Planck scale may be effectively two-dimensional, while the latter implies a condition of maximal compression of matter by the formation of an event horizon for Planckian scattering. We generalize such a result to an arbitrary number of dimensions, and show that gravity in higher than four dimensions remains self-complete, but in lower dimensions it is not. In such a way we established an "exclusive disjunction" or "exclusive or" (XOR) between the occurrence of self-completeness and dimensional reduction, with the goal of actually reducing the unknowns for the scenario of the physics at the Planck scale. Potential phenomenological implications of this result are considered by studying the case of a two-dimensional dilaton gravity model resulting from dimensional reduction of Einstein gravity.