Generalized RG Approach

Abstract

In Chapters 3 and 4, two ideologically different RG approaches have been considered. In Wilson’s approach one deals with the Hamiltonian Eq. (3.24) which includes all imaginable vertexes complying with the symmetry of the problem. By contrast the field theoretical RG can only be effectively used with the Hamiltonian containing a small number of vertexes which do not violate renormalizability of the problem. Notwithstanding such a substantial difference in the formulation, the results of calculations in these approaches in the critical region are practically the same. Therefore, there is some property, common to both approaches, which is of crucial importance for the description of critical region. This property is Kadanoff’s invariance completing the calculational schemes in both methods. Kadanoff’s invariance is an intrinsic property of a system which can be found in a particular point of the parametric space of the Hamiltonian describing this system. This point may or may not exist, but the scale transformations and the consequent scale covariance is always present in any physical problem being an inherent attribute of the configurational space. Hence, one could start the study of a system from the scale transformations and then, looking for the point of Kadanoff’s invariance, come to the origin from where these two different approaches merge.