泛函积分方法求解Maxiko-Migdal方程
A Functional Integral Approaches to the Makeenko-Migdal Equations
摘要：
Maurko和Migdal（1979）给出了启发式恒等式，涉及到与在自交点处分裂单个循环相关联的两个威尔逊圈泛函的乘积的期望。卡扎科夫和我。K. Kostov（1980）将平面情况下的Majuko-Migdal方程重新表述为严格意义上的形式。然而，直到T。Lévy（2011年）。随后Driver，Kemp和Hall（2017）给出了Lévy结果的简化证明，然后用F. Gabriel（2017）证明了这些简化的证明可以扩展到任意紧曲面上的Yang-Mills测度。到目前为止，所有的证明都是有限维部分积分的基本但棘手的练习。本文的目的是给出严格的功能积分证明的Mauryko-Migdal方程的指导下，原来的启发式机器发明的Mauryko和Migdal。虽然这种随机证明在技术上更困难，但它在概念上更清晰，并解释了“为什么”Mauryko-Migdal方程是正确的。我们希望本文也将作为一个介绍的一些问题所涉及的意义量化杨密尔的领域。
英文摘要：
Makeenko and Migdal (1979) gave heuristic identities involving the expectation of the product of two Wilson loop functionals associated to splitting a single loop at a self-intersection point. Kazakov and I. K. Kostov (1980) reformulated the Makeenko--Migdal equations in the plane case into a form which made rigorous sense. Nevertheless, the first rigorous proof of these equations (and their generalizations) was not given until the fundamental paper of T. Lévy (2011). Subsequently Driver, Kemp, and Hall (2017) gave a simplified proof of Lévy's result and then with F. Gabriel (2017) we showed that these simplified proofs extend to the Yang-Mills measure over arbitrary compact surfaces. All of the proofs to date are elementary but tricky exercises in finite dimensional integration by parts. The goal of this article is to give a rigorous functional integral proof of the Makeenko--Migdal equations guided by the original heuristic machinery invented by Makeenko and Migdal. Although this stochastic proof is technically more difficult, it is conceptually clearer and explains "why" the Makeenko--Migdal equations are true. It is hoped that this paper will also serve as an introduction to some of the problems involved in making sense of quantizing Yang-Mill's fields.
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