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.

我们研究了几个非微扰二维量子引力的玩具模型中近似杀伤向量场的概念。使用离散外部微积分的框架，我们展示了如何制定与这种近似杀死矢量场相关的量子可观测量。使用这些方法，我们的目标是研究由手头的量子引力模型产生的时空几何的对称性。由于我们期望量子涨落在小尺度上占主导地位，因此我们的目标是构建一个与尺度相关的对称性概念，该概念可用于确定涌现（半）经典几何是否承认任何近似的 Killing 对称性。我们已经在离散几何的三个集合上评估了这样一个可观测量的一个特定选择。我们发现该方法在波动较小的情况下很有用，