标题： 在非交换几何中错失要点 显示英文标题

标题： Missing the point in noncommutative geometry

摘要： 非交换几何推广了标准的光滑几何，用一个具有面积量纲的基本量来参数化维度的非交换性。 于是就出现了这样的问题：在这样的理论中，比尺度更小的区域的概念——最终是点的概念——是否成立。 我们认为，从两个相互关联的角度来看，答案是否定的。 在康尼斯的谱三元组方法的背景下，我们证明了任意小的区域在形式上是无法定义的。 而在标量场莫尔-维勒方法中，我们证明了它们无法被赋予操作性定义。 因此我们得出结论，这样的几何中不存在点。 因此，我们研究了(a)这种几何的形而上学，以及(b)如何将平滑流形的表象作为对基本非交换几何的近似来恢复。 显示英文摘要

摘要： Noncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale - and ultimately the concept of a point - makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes' spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal-Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.