非对易几何中的失点
Missing the point in noncommutative geometry
摘要：
非对易几何推广了标准光滑几何，用面积维数作为基本量来参数化维数的非对易性。于是问题就出现了，在这样一个理论中，一个比尺度小的区域的概念--以及最终一个点的概念--是否有意义。我们认为，它没有，在两个相互关联的方式。在Connes的谱三元组方法的背景下，我们表明，任意小的区域是不能定义的正式意义上。而在标量场Moyal-Weyl方法，我们表明，他们不能给出一个操作的定义。我们的结论是，点不存在于这样的几何形状。因此，我们研究（a）这样的几何形而上学，（b）光滑流形的外观如何可能被恢复为一个基本的非交换几何的近似。
英文摘要：
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.
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