Wilson Renormalization Group and Realization of Symmetries in Field Theories

威尔逊重正化群和场论中对称性的实现

基本信息

批准号：
13235209
负责人：
IGARASHI Yuji
金额：
--
依托单位：
Niigata University
依托单位国家：
日本
项目类别：
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13235209/
关键词：
Field Theory Renormalization Group Gauge Symmetry Chiral Symmetry Lattice Theories Quantum Master Equation

项目摘要

The Wilsonian reanormalization group is known to employ a momentum cut-off in order to perform successive integration of quantum modes to define an effective action. This apparently violates gauge symmetries. The purpose of this research project is to realize such symmetries, which are naively incompatible with the given regularization scheme. We study new realization of symmetries for these 6 years not only in continuum theories but also in lattice theories, chiral symmetry, for example. Our main machinery to attack the problem is "antifield formalism" which provides with most general formulation of local as well as global symmetries.We apply(1) the antifield formalism to the Wilsonian action in renormalization group(2) the block transformation corresponding to coarse graining, which relates UV fields to IR fields to realize new kind of symmetries which are naively incompatible with underlying regularization scheme. We find this kind of symmetries can be described by the Quantum Master Equation in the antifield formalism. We have shown that if the Quantum Master Equation holds for a cut-off removed theory, it is also the case for the theory with a finite cut-off. Thereby, we have established at least conceptually the presence of effective but exact symmetries. It is hard, however, to set up the Quantum Master Equation and to construct its solutions for given system with symmetry specified for cut-off removed actions. First, we have investigated lattice chiral symmetry, and shown that the Gisnparg-Wilson relation, the Ward-Takahashi identity, which plays the crucial role for describing new type of lattice chiral symmetry is nothing but the Quantum Master Equation. We also solved the Quantum Master Equation for a fermionic interacting theory.In the last year, we study new realization of gauge symmetry in the Wilson renormalization group, and gave the Quantum Master Equation and its formal solution for QED.

众所周知，威尔逊重整化群采用动量截止来执行量子模式的连续积分以定义有效的作用。这显然违反了规范对称性。该研究项目的目的是实现这种对称性，这与给定的正则化方案天真地不兼容。这六年来，我们不仅在连续介质理论中研究对称性的新认识，还在格子理论中研究对称性，例如手性对称性。我们解决这个问题的主要机制是“反场形式主义”，它提供了局部和全局对称性的最一般的表述。我们将（1）反场形式主义应用于重正化群中的威尔逊作用（2）对应于粗略的块变换颗粒化，它将 UV 场与 IR 场联系起来，以实现与底层正则化方案天真不兼容的新型对称性。我们发现这种对称性可以用反场形式主义中的量子主方程来描述。我们已经证明，如果量子主方程对于去除截止的理论成立，那么对于具有有限截止的理论也是如此。因此，我们至少在概念上确立了有效但精确的对称性的存在。然而，建立量子主方程并为给定系统构造其解（具有指定截止移除作用的对称性）是很困难的。首先，我们研究了晶格手性对称性，并表明在描述新型晶格手性对称性中起关键作用的Gisnparg-Wilson关系、Ward-Takahashi恒等式正是量子大师方程。我们还求解了费米相互作用理论的量子主方程。去年，我们研究了威尔逊重正化群中规范对称性的新实现，给出了QED的量子主方程及其形式化解。