Renormalization for quantum systems

The quantum nature of physical systems shows up at very low temperatures and affects phase transition phenomena. The existence of quantum transitions at T = 0 leads to the manifestation of quantum-classical crossover phenomena in classical transitions at low temperatures. The real space renormalization group methods introduced for classical systems have been extended to quantum systems at T not= 0. At T = 0 a block renormalization group method has been introduced to study the ground state and the excited states of a many-body quantum system and to analyze transitions which take place in the ground state of the system. These different methods are described and discussed, giving more emphasis to the block method. The numerous applications to spin and fermion systems and to field theory in one and higher dimensions are presented. (orig.)