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arXiv 1003.1366hep-thcond-mat.stat-mechcond-mat.str-elhep-ph
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正合重整化群的基本原理

Fundamentals of the Exact Renormalization Group

Oliver J. Rosten

备注：
172 pages, 9 figures; v2: 177 pages, corrections and improvements (particularly to the discussion of lines of equivalent fixed-points) + refs added; v3: 180 pages, small refinements and some minor corrections; v4: 198 pages, 10 figures, Published in Phys. Repts. - significant additional material esp. on the effective average action, optimization and gauge theory
发表信息：
Physics Reports 511 (2012), pp. 177-272
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摘要：

精确重整化群（ERG）的各个方面进行了探讨，开始审查的概念支撑的框架和情况下，它预计是有用的。一个特别强调的是放在直观的图片上提供的重整化在量子场论和普遍性与二阶相变。对平凡性、渐近自由性和渐近安全性进行了定性讨论。聚焦于标量场理论，被认为是使用一般的方法，从而不同的ERG遵循从字段重新定义的各种流量方程的建设。我们记得波尔钦斯基方程可以被转换为热方程，这为下面的内容提供了直观和计算技术。流动方程精确解的性质分析包括证明在临界不动点处反常维数的谱是量子化的。考虑了计算λ φ ^4理论中β函数的两种替代方法。对于其中之一，它被发现，所有明确的依赖于一个家庭的ERG之间的非普遍的差异抵消，正是。Wilson-Fisher不动点以一种相当新颖的方式被重新发现。非微扰近似方案的讨论集中在衍生物的扩展，并包括一个完善的参数，在最低阶在这个近似，可以构造一个函数沿流动单调递减。对Wilson有效作用量的可重整化性与关联函数的关系提供了一个新的视角，并在此基础上勾画了明显规范不变ERG的构造，给出了一些新的见解.把这些线索联系在一起，提出了一种研究量子场论的新方法。

英文摘要：

Various aspects of the Exact Renormalization Group (ERG) are explored, starting with a review of the concepts underpinning the framework and the circumstances under which it is expected to be useful. A particular emphasis is placed on the intuitive picture provided for both renormalization in quantum field theory and universality associated with second order phase transitions. A qualitative discussion of triviality, asymptotic freedom and asymptotic safety is presented. Focusing on scalar field theory, the construction of assorted flow equations is considered using a general approach, whereby different ERGs follow from field redefinitions. It is recalled that Polchinski's equation can be cast as a heat equation, which provides intuition and computational techniques for what follows. The analysis of properties of exact solutions to flow equations includes a proof that the spectrum of the anomalous dimension at critical fixed-points is quantized. Two alternative methods for computing the beta-function in lambda phi^4 theory are considered. For one of these it is found that all explicit dependence on the non-universal differences between a family of ERGs cancels out, exactly. The Wilson-Fisher fixed-point is rediscovered in a rather novel way. The discussion of nonperturbative approximation schemes focuses on the derivative expansion, and includes a refinement of the arguments that, at the lowest order in this approximation, a function can be constructed which decreases monotonically along the flow. A new perspective is provided on the relationship between the renormalizability of the Wilsonian effective action and of correlation functions, following which the construction of manifestly gauge invariant ERGs is sketched, and some new insights are given. Drawing these strands together suggests a new approach to quantum field theory.

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