Wilson Renormalization Group and Realization of Symmetries in Field Theories

众所周知，威尔逊重归一化群使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。本研究项目的目的是实现这种与给定正则化方案天真不相容的对称性。在这6年中，我们不仅在连续统理论中，而且在晶格理论中，例如手征对称性，研究了对称性的新实现。我们解决这个问题的主要机制是“反场形式”，它提供了最普遍的局部和全局对称性的表述。我们将(1)反场形式应用于重整化群中的威尔逊作用(2)与粗粒化相对应的块变换，它将紫外场与红外场联系起来，以实现与基本正则化方案天真不相容的新型对称性。我们发现这种对称性可以用反场形式中的量子主方程来描述。我们已经证明，如果量子主方程对截断去掉的理论成立，那么它也适用于有限截断的理论。因此，我们至少在概念上确定了有效但精确的对称性的存在。然而，要建立量子主方程并构造它的解是困难的，因为给定的系统具有指定的截断移除作用的对称性。首先，我们研究了晶格手征对称性，证明了对描述新型晶格手征对称性起关键作用的Gisnparg-Wilson关系、Ward-Takahashi恒等式就是量子主方程。在过去的一年里，我们研究了规范对称性在威尔逊重整化群中的新实现，给出了量子电动力学的量子主方程及其形式解。

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.