Quantum Action Principle and Ward Identities

Abstract

The symmetries of a quantum field model are expressed by Ward identities, i.e., by certain identities which the Green functions of the theory must obey. The aim of the renormalization procedure is to prove the renormalizability of a given symmetry of the corresponding classical theory. This means that one has to construct a theory of Green functions fulfilling perturbatively, to all orders, the Ward identities which express this symmetry. We shall explain in this chapter how to write the Ward identities in the functional formalism, beginning with the classical theory — equivalent to the tree approximation of the quantum theory — and then we shall proceed to the extension to all orders of perturbation theory, for very simple models with rigid symmetry. The main tool being the renormalized [12, 13, 14] version of the Schwinger action principle [56], the quantum action principle (QAP), a section will be devoted to the latter.