Title: Lectures on the functional renormalization group method Show Chinese title

Title: 关于功能重整化群方法的讲座

Abstract: These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained in a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scalar theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED. Show Chinese abstract

Abstract: 这些简介性笔记涉及功能重整化群方程及其一些应用。 强调此方法的应用范围远远超出临界系统，实际上为我们提供了一个通用算法来求解强耦合量子场论。 F. Wegner 和 A. Houghton 的重整化群方程被证明可以重求和环展开。 另一种版本由 J. Polchinski 提出，是通过集体坐标的方法得到的，可用于微扰级数的重求和。 真正非微扰的演化方程以类似于 Schwinger-Dyson 方程的方式得到。 介绍了该方案的两种变体，其中决定依次消除模式的尺度分别从外部空间和内部空间中提取。 简要讨论了复合算符的重整化作为到达重整化群方程的另一种方式。 从局部和全局的观点考虑了标度定律和固定点。 在标量理论的对称性破缺相中，展示了由不稳定性引起的重整化和新的标度定律。 在 sine-Gordon 模型的情况下，展示了紧致变量的有效势能的平坦化。 最后，给出了 QED 的显式规范不变演化方程。