Brief Summary of Each Supplement
Progress of Theoretical Physics Supplement No. 181
Realization of Symmetry in the ERG Approach to Quantum Field Theory
By Y. Igarashi, K. Itoh and H. Sonoda
The exact renormalization group (ERG) has been applied for many years
to a variety of fields, including field theory and critical
phenomena. It has gained the reputation
as a practical method of non-perturbative approximations.
In this review we explain the ERG formulation of field theory
emphasizing the following two aspects:
1)   how to construct the continuum limit of a field theory,
2)   how to introduce continuous symmetry.
We complement the general theory with many but mostly perturbative
In the ERG formulation of field theory, a theory is defined through
the Wilson action or equivalently the effective average action.
We first introduce the two types of actions and explain their relationship.
In this review we mainly discuss the Wilson action because of the relative
ease of incorporating symmetry with it.
We then proceed to such topics as renormalizability, continuum
limits, and "composite operators"; the last of which are defined via flow
equations and play an important role in the realization of symmetry.
Ordinarily, regularization of a field theory with a momentum
cutoff may conflict with the symmetry of the theory, especially local
gauge symmetry. Using ERG, however, any continuous symmetry can be realized
with no compromise. This situation is analogous to the realization of chiral
symmetry on a lattice. We devote the second half of the review to the
realization of continuous symmetry via the Ward-Takahashi identity or
the quantum master equation in the antifield formalism. We elucidate
the general theory with concrete examples.
We have written the review with a sincere hope that the exact
renormalization group would become part of the shared knowledge among
all the practitioners of field theory.