Quantum Gravity in Large Dimensions

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 $k_c/\lambda=1/d$ (with $k=1/8 \pi G$) 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 $\nu=0$ at $d=\infty$.