We will consider solutions of three physically important problems by the renormalization group methods: i) Perturbative renormalization of SU(2) Yang-Mills. The main part of the work is to prove at all loop orders the existence of the vertex functions and the restoration of the Slavnov-Taylor identities in the renormalized theory. ii) Dynamical symmetry breaking in su(2) Yang-Mills. We make progress towards a derivation of a low energy effective theory for SU(2) Yang–Mills theory. This low energy action is computed to 1-loop using the renormalization group technique. After that, we perform the Spin-Charge decomposition in a way proposed by L.D. Faddeev and A.J. Niemi. The effective potential is of Mexican hat type. iii) Lamb shift in heavy atoms. We formulate the Hartree–Fock method in the contemporary language of QED using a functional integral approach. This allows us to set up the corresponding renormalization group equation which can be solved numerically in the large-N approximation.
