Symmetry properties, ward identities and renormalization group for Fermi and Bose systems

Within the framework of the perturbative renormalization-group approach, we discuss the case when perturbation theory is affected by infrared divergences well inside a stable phase of matter. Contrary to the case of critical phenomena, a cancellation of the singularities, rather than a resummation, has to take place, in order to obtain finite response functions, as implied by the stability conditions. We show how symmetries, and related Ward identities, can be implemented to relate the singular parts of the different running coupling constants, and reduce the number of independent parameters, thus allowing for a closure of the renormalization-group equations, and for the asymptotical solution of the problem. Applications to the cases of Fermi and Bose systems in low spatial dimensions are discussed.