Summing Logarithms in Quantum Field Theory: The Renormalization Group

Abstract

The process of renormalization in quantum fieldtheory necessarily involves the introduction of anarbitrary mass scale μ2 into the theory.The effect of having this parameter appear due toquantum effects can be analyzed from many points of view; thegeneral topic is usually called the“renormalization group.” In this paper, oneaspect of this feature of quantum field theory isdiscussed in some detail. It is shown how the appearance of thisarbitrary mass scale imposes consistency conditions onquantum-induced corrections to the classical action ofa model. This has the effect of determining higher order corrections in terms of lower ordercorrections in the perturbative expansion of theeffective action, which in turn permits at least partialsummation of all terms in the perturbative expansion. This is illustrated in the context of twosimple, well-understood models; a φ4model in four dimensions and a φ3 modelin six dimensions. The technicalities associated withthe renormalization procedure itself are not discussed.