We explain qualitatively in this short paper the essence of the renormalization group, showing that these ideas apply not only in physics but also beyond, forming the foundation of artificial intelligence.
We start with a simple introduction into the renormalization group (RG) in quantum field theory and give an overview of the renormalization group method. The third section is devoted to essential topics of the renorm-group use in the QFT. Here, some fresh results are included. Then we turn to the remarkable proliferation of the RG ideas into various fields of physics. The last section summarizes an impressive recent progress of the "QFT renormalization group" application in mathematical physics.
These lecture notes have been written for a short introductory course on universality and renormalization group techniques given at the VIII Modave School in Mathematical Physics by the author, intended for PhD students and researchers new to these topics.First the basic ideas of dynamical systems (fixed points, stability, etc.) are recalled, and an example of universality is discussed in this context: this is Feigenbaum's universality of the period doubling cascade for iterated maps on the interval.It is shown how renormalization ideas can be applied to explain universality and compute Feigenbaum's constants.Then, universality is presented in the scenario of quantum field theories, and studied by means of functional renormalization group equations, which allow for a close comparison with the case of dynamical systems.In particular, Wetterich equation for a scalar field is derived and discussed, and then applied to the computation of the Wilson-Fisher fixed point and critical exponent for the Ising universality class.References to more advanced topics and applications are provided.
This review covers several topics involving renormalization group ideas. The solution of the s-wave Kondo Hamiltonian, describing a single magnetic impurity in a nonmagnetic metal, is explained in detail. See Secs. VII-IX. "Block spin" methods, applied to the two dimensional Ising model, are explained in Sec. VI. The first three sections give a relatively short review of basic renormalization group ideas, mainly in the context of critical phenomena. The relationship of the modern renormalization group to the older problems of divergences in statistical mechanics and field theory and field theoretic renormalization is discussed in Sec. IV. In Sec. V the special case of "marginal variables" is discussed in detail, along with the relationship of the modern renormalization group to its original formulation by Gell-Mann and Low and others.