关于规范理论在't Hooft极限的涌现
On emergence in gauge theories at the 't Hooft limit
摘要：
本文的目的是通过讨论一个著名的物理极限--特胡夫极限，有助于更好地概念理解规范量子场论，如量子色动力学。极限的概念是颜色（或电荷）的数量$N$趋于无穷大。在这个极限下可能发生的简化，我们将考虑的是：（1）理论的费曼图可以画在一个没有相交线的平面上（称为“平面性”）;（2）理论，或它的一个扇区，变得可积，并且确实对应于一个研究得很好的系统，即自旋链。平面性很重要，因为它表明了量子场论如何表现出扩展的、特别是类弦的结构;在某些情况下，这与弦论，从而与引力有联系。以前关于一个理论（或一个理论的一个部门或体系）如何从另一个理论中产生或还原为另一个理论的哲学文献倾向于强调一些情况，比如在统计力学中，系统在极限之前有200多个自由度。但在这里，我们的量子场论，包括那些在通往特胡夫特极限的路上的理论，将有无限多个自由度。然而，我们将说明巴特菲尔德最近提出的一个模式和诺顿的分类法如何适用于我们所考虑的量子场论;我们将用这些术语对我们的理论的三个物理性质进行分类。这些性质是平面性和可积性，如上面的（i）和（ii）;以及β函数的行为，例如，渐近自由。我们对这些性质的讨论，尤其是对β函数的讨论，也将涉及到最近关于评估量子场论是否合适的哲学辩论，量子场论的严格存在尚未得到证明。
英文摘要：
The aim of this paper is to contribute to a better conceptual understanding of gauge quantum field theories, such as quantum chromodynamics, by discussing a famous physical limit, the 't Hooft limit, in which the theory concerned often simplifies. The idea of the limit is that the number $N$ of colours (or charges) goes to infinity. The simplifications that can happen in this limit, and that we will consider, are: (i) the theory's Feynman diagrams can be drawn on a plane without lines intersecting (called `planarity'); and (ii) the theory, or a sector of it, becomes integrable, and indeed corresponds to a well-studied system, viz. a spin chain. Planarity is important because it shows how a quantum field theory can exhibit extended, in particular string-like, structures; in some cases, this gives a connection with string theory, and thus with gravity. Previous philosophical literature about how one theory (or a sector, or regime, of a theory) might be emergent from, and-or reduced to, another one has tended to emphasize cases, such as occur in statistical mechanics, where the system before the limit has finitely many degrees of freedom. But here, our quantum field theories, including those on the way to the 't Hooft limit, will have infinitely many degrees of freedom. Nevertheless, we will show how a recent schema by Butterfield and taxonomy by Norton apply to the quantum field theories we consider; and we will classify three physical properties of our theories in these terms. These properties are planarity and integrability, as in (i) and (ii) above; and the behaviour of the beta-function reflecting, for example, asymptotic freedom. Our discussion of these properties, especially the beta-function, will also relate to recent philosophical debate about the propriety of assessing quantum field theories, whose rigorous existence is not yet proven.
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