Approximate Killing symmetries in non-perturbative quantum gravity

Abstract

We study the notion of approximate Killing vector fields in several toy models of non-perturbative two-dimensional quantum gravity. Using the framework of discrete exterior calculus, we show how to formulate quantum observables related to such approximate Killing vector fields. Using these methods, we aim to investigate symmetry properties of the space–time geometry produced by the quantum gravitational model at hand. Since we expect quantum fluctuations to dominate at small scales, our goal is to construct a scale-dependent notion of symmetry that might be used to determine whether the emergent (semi-)classical geometry admits any approximate Killing symmetries. We have evaluated one particular choice of such an observable on three ensembles of discrete geometry. We find that the method is useful in the setting where fluctuations are small, but that more work is needed before these ideas can be applied in the deep quantum regime.