Parisi-Sourlas-like dimensional reduction of quantum gravity in the presence of observers

Abstract

One of the sources of incompatibility between general relativity and quantum mechanics is perturbative non-renormalizability of quantum gravity in $3+1$ spacetime dimensions. Here, we show that in the presence of disorder induced by random networks of observers measuring covariant quantities (such as scalar curvature) $(3+1)$-dimensional quantum gravity exhibits an effective dimensional reduction at large spatio-temporal scales, which is analogous to the Parisi-Sourlas phenomenon observed for quantum field theories in random external fields. After averaging over associated disorder and focusing on the infrared dynamics of the theory we find that the upper critical dimension of quantum gravity is lifted from $D_{{\rm cr}}=1+1$ to $D_{{\rm cr}}=3+1$ dimensions.