单点紧致化

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 2.3 拓扑空间

读者或许无法从定义和这里给出的少数几个例子中体会到紧致性的意义。然而应当注意，某些数学分析以及物理学在紧致空间上会变得相当简单。例如，让我们考虑固体中的电子系统。如果固体是非紧致的、具有无限体积，我们就必须在无限体积中处理量子统计力学。众所周知，这在数学上相当复杂，需要用到希尔伯特空间的高级理论。我们通常的做法是把系统限制在被硬壁包围的有限体积 $V$ 中，使电子波函数在壁上为零；或者在壁上施加周期性边界条件，这相当于把系统放在一个环面上，见例 2.5(b)。无论哪种情形，系统现在都被置于一个紧致空间中。于是我们可以构造其激发由离散指标标记的福克空间。紧致性在物理学中的另一重要意义将在我们研究瞬子和Belavin-Polyakov 单极子这类延展对象时显现，见4.8 节。在场论中，我们通常假定场在空间无穷远处趋近于某个对应于真空态（或其中一个真空态）的渐近形式。类似地，空间无穷远处具有共同序参量的一类序参量分布，从各种观点来看都是值得研究的有趣类别，这一点我们将在后面看到。由于无穷远处的所有点都被映到同一个点，我们实际上已把非紧致空间 ${\mathbb{R}}^n$ 紧化为紧致空间 $S^n={\mathbb{R}}^n\cup \{\infty \}$。这一过程称为单点紧致化(one-point compactification)。

英文原文

The reader might not appreciate the significance of compactness from the definition and the few examples given here. It should be noted, however, that some mathematical analyses as well as physics become rather simple on a compact space. For example, let us consider a system of electrons in a solid. If the solid is non-compact with infinite volume, we have to deal with quantum statistical mechanics in an infinite volume. It is known that this is mathematically quite complicated and requires knowledge of the advanced theory of Hilbert spaces. What we usually do is to confine the system in a finite volume $V$ surrounded by hard walls so that the electron wavefunction vanishes at the walls, or to impose periodic boundary conditions on the walls, which amounts to putting the system in a torus, see example 2.5(b). In any case, the system is now put in a compact space. Then we may construct the Fock space whose excitations are labelled by discrete indices. Another significance of compactness in physics will be found when we study extended objects such as instantons and Belavin-Polyakov monopoles, see section 4.8. In field theories, we usually assume that the field approaches some asymptotic form corresponding to the vacuum (or one of the vacua) at spatial infinities. Similarly, a class of order parameter distributions in which the spatial infinities have a common order parameter is an interesting class to study from various points of view as we shall see later. Since all points at infinity are mapped to a point, we have effectively compactified the non-compact space ${\mathbb{R}}^n$ to a compact space $S^n={\mathbb{R}}^n\cup \{\infty \}$ . This procedure is called the one-point compactification.

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