标题： 量子引力：我们到了吗？ 显示英文标题

标题： Quantum Gravity: are we there yet?

摘要： 千禧年之际，人们对一种受称为“量子群”的丰富数学对象启发的非交换几何方法充满乐观，并将其应用于量子时空。这将把量子引力效应建模为时空坐标的非交换性，并可能解决量子引力本身。从那时起，又花了20年时间来发展一种特别合适的“量子黎曼几何”形式，但这一工作已经基本完成，并开始用于构建婴儿量子引力模型。在本文中，我们获得了该方法中最新模糊球体和n边形模型的新结果。我们还回顾了目前已能看到的量子引力的一些要素，以及为了更全面地实现这一目标仍缺失的关键概念和数学要素。 显示英文摘要

摘要： The turn of the millennium was a time of optimism about an approach to noncommutative geometry inspired by rich mathematical objects called `quantum groups' and its applications to quantum spacetime. This would model quantum gravity effects as noncommutativity of spacetime coordinates and was arguably going to solve quantum gravity itself. It took a further 20 years from that point to develop a particularly suitable formalism of `quantum Riemannian geometry', but this was largely done and has begun to be used to construct baby quantum gravity models. In this article, we obtain new results for state of the art fuzzy sphere and n-gon models in this approach. We also review what are some elements of quantum gravity that we can already see and what are the critical conceptual and mathematical elements that are still missing to more fully achieve this goal.