Renormalization Group Solution of Fermionic Dyson Model

Abstract

In this paper we give a brief exposition of the fermionic version of Dyson’s hierarchical model. Action of Wilson’s renormalization group (RG) transformation in the space of coupling constants of hierarchical fermionic model is given by the rational map. Global RG-flow in the upper half-plane of the coupling constants is described. Complex behaviour of stable RG-invariant curves leads to the non-trivial picture of critical phenomena in this model. The continuum limit of the hierarchical fermionic model is given by p-adic fermionic model. The connection between coupling constants of p-adic model and its discretized hierarchical version is defined by discretization operator, which is given by non-trivial functional integral. This operator can be considered as a normalizing transformation for RG-map at trivial fixed point. The ultraviolet poles of Feynman amplitudes appear as a resonance values for normalizing transformation. Convergence of functional integral follows from Poincare and Siegel theorems.

This work was supported in part by the Russian Foundation for Basic Research (Grants No.99-01-00467 and No.01-01-00610).