We propose an approach which, by combining insights from loop quantum gravity (LQG), Topos theory, non-commutative geometry la Connes, and spacetime relationalism, provides fertile ground for the search of an adequate spacetime picture in quantum gravity. With this approach, we obtain a novel way of deducing the quantization of the possible values for the area of a surface. One gets the same area values than those from the area operator in standard LQG, but our approach makes a further prediction: some smaller values and sub-divisions are also allowed. In addition, the area arises as a noncommutative distance between two noncommutative points, and thus they should be interpreted as irreducible string-like objects at the physical level (where the area interpretation for the noncommutative distance holds).

我们提出了一种方法，该方法结合了环量子引力 (LQG)、拓扑理论、非交换几何la Connes 和时空关系论的见解，为在量子引力中寻找足够的时空图提供了肥沃的土壤。通过这种方法，我们获得了一种推导表面面积可能值量化的新方法。与标准 LQG 中的面积算子获得的面积值相同，但我们的方法进行了进一步的预测：一些较小的值和细分也是允许的。此外，该区域作为非交换距离出现 在两个非对易点之间，因此它们应该被解释为物理水平上不可约的弦状物体（非对易距离的面积解释成立）。