标题： 线性化重正化 显示英文标题

标题： Linearized renormalization

摘要： 利用无穷小方法，本文研究了重整化问题以处理量子场论中出现的紫外发散。 假设作用量已经经过重整化，从而得到满足某些重整化条件的超有限有效作用量，我们分析了作用量必须如何调整以重现这些重整化条件的一阶变化。 然后，该分析提供了理论相关函数所受的变化。 在这个程序中，对于超可重整理论（即四维时空中的具有三次相互作用的标量场和三维时空中的具有四次相互作用的标量场）取得了成功。 基于微扰重整化理论中的现有结果，我们为这些理论推导出显式的重整化表达式，每个表达式仅涉及由全传播子和全$n$-点顶点构造的有限数量的图。 可重整情况也进行了分析；当去除调节器时，所得表达式是超有限的，但不能在没有调节器的情况下写出。 从这个意义上说，在可重整情况下，重整化并不是完全显式的。 然而，从自由理论开始的方程的微扰解提供了重整化的费曼图，类似于BPHZ程序。 为了与保持重整化条件相兼容，引入了一种投影重整化方案，而不是最小方案。 所发展的思想被扩展到复合算符的重整化以及施温格-狄拉克方程的研究中。 显示英文摘要

摘要： Using an infinitesimal approach, this work addresses the renormalization problem to deal with the ultraviolet divergences arising in quantum field theory. Under the assumption that the action has already been renormalized to yield an ultraviolet-finite effective action that satisfies a certain set of renormalization conditions, we analyze how the action must be adjusted to reproduce a first-order change in these renormalization conditions. The analysis then provides the change that is induced on the correlation functions of the theory. This program is successfully carried out in the case of super-renormalizable theories, namely, a scalar field with cubic interaction in four space-time dimensions and with quartic interaction in three space-time dimensions. Relying on existing results in the theory of perturbative renormalization, we derive explicit renormalized expressions for these theories, each of which involves only a finite number of graphs constructed with full propagators and full $n$-point vertices. The renormalizable case is analyzed as well; the derived expressions are ultraviolet finite as the regulator is removed but cannot be written without a regulator. In this sense, the renormalization is not fully explicit in the renormalizable case. Nevertheless, a perturbative solution of the equations starting from the free theory provides the renormalized Feynman graphs, similar to the BPHZ program. For compatibility with the preservation of the renormalization conditions, a projective renormalization scheme, as opposed to a minimal one, is also introduced. The ideas developed are extended to the study of the renormalization of composite operators and the Schwinger-Dyson equations.