Quantum Gravity in Large Dimensions

TL;DR

This work develops two nonperturbative, large-d approaches to quantum gravity using Regge lattice gravity: a weak-field 1/d expansion on a cross-polytope lattice and a strong-coupling, large-d expansion tied to random-surface geometry. It identifies a lattice critical point at k_c proportional to λ/d^3 (with k=1/8πG), accompanied by a large number of zero modes, and shows curvature correlations are governed by large, essentially unconstrained random surfaces, yielding a gravitational correlation length ξ that scales as |log(k_c − k)|^{1/2} and a universal exponent ν = 0 in the d → ∞ limit. The study also connects lattice results to continuum expectations, examining Feynman rules and dimensional dependence in d dimensions, and discusses the implications for nonperturbative quantum gravity in four dimensions. Overall, the paper provides a controlled nonperturbative lattice perspective on quantum gravity’s high-dimensional limit and offers insight into how nontrivial curvature scales and RG running might emerge in the continuum limit.

Abstract

Quantum gravity is investigated in the limit of a large number of space-time dimensions, using as an ultraviolet regularization the simplicial lattice path integral formulation. In the weak field limit the appropriate expansion parameter is determined to be $1/d$. For the case of a simplicial lattice dual to a hypercube, the critical point is found at $k_c/λ=1/d$ (with $k=1/8 πG$) separating a weak coupling from a strong coupling phase, and with $2 d^2$ degenerate zero modes at $k_c$. The strong coupling, large $G$, phase is then investigated by analyzing the general structure of the strong coupling expansion in the large $d$ limit. Dominant contributions to the curvature correlation functions are described by large closed random polygonal surfaces, for which excluded volume effects can be neglected at large $d$, and whose geometry we argue can be approximated by unconstrained random surfaces in this limit. In large dimensions the gravitational correlation length is then found to behave as $| \log (k_c - k) |^{1/2}$, implying for the universal gravitational critical exponent the value $ν=0$ at $d=\infty$.