Wilson Renormalization Group and Realization of Symmetries in Field Theories

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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.