Most quantum field theories (QFT) of physical interest exhibit symmetries, exact symmetries or symmetries with soft (e.g. linear) breaking. This chapter deals only with linear continuous symmetries corresponding to compact Lie groups. When the bare action has symmetry properties, to preserve the symmetry it is first necessary to find a symmetric regularization. The symmetry properties of the QFT then imply relations between connected correlation functions, and vertex functions, called Ward–Takahashi (WT) identities, which describe the physical consequences of the symmetry. WT identities also constrain UV divergences, and the counter-terms that render the theory finite are not of most general form allowed by power counting. As a consequence the renormalized action is expected to keep some trace of the initial symmetry. Such an analysis is based on a perturbative loop expansion. More generally, some non-trivial relations survive when to the action are added terms that induce a soft breaking of symmetry (i.e. by relevant terms). The specific examples of linear symmetry breaking, and the very important limiting case of spontaneous symmetry breaking, and quadratic symmetry breaking are examined. Finally, as an application, the example of chiral symmetry breaking in low-energy effective models of hadron physics is discussed.
Canonical brackets of a toy model for the Hodge theory without its canonical conjugate momenta
