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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 examples.

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.

  Copyright © 2012 Progress of Theoretical Physics