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.