Realization of symmetries and the c – theorem

We discuss the relation between the c–theorem and the the way various symmetries are realized in quantum field theory. We review our recent proof of the c–theorem in four dimensions. Based on this proof and further evidence, we conjecture that the realization of chiral symmetry be irreversible, flowing from the Wigner-Weyl realization at short distance to the Nambu–Goldstone realization at long distance. We argue that three apparently independent constraints based on the renormalization group, namely anomaly matching, the c–theorem and the conjectured symmetry realization theorem, are particular manifestations of a single underlying principle. Presented by J.I.L. at the Faro Workshop on Exact Renormalization Group September 10-12, 1998. to be published in the proceedings