Scale-Dependent Functions, Stochastic Quantization and Renormalization

Abstract

We consider a possibility to unify the methods of regularization, such as the renormalization group method, stochastic quantization etc., by the extension of the standard field theory of the square-integrable functions $ϕ(b)\in L^2({\mathbb R}^d)$ to the theory of functions that depend on coordinate $b$ and resolution $a$. In the simplest case such field theory turns out to be a theory of fields $ϕ_a(b,\cdot)$ defined on the affine group $G:x'=ax+b$, $a>0,x,b\in {\mathbb R}^d$, which consists of dilations and translation of Euclidean space. The fields $ϕ_a(b,\cdot)$ are constructed using the continuous wavelet transform. The parameters of the theory can explicitly depend on the resolution $a$. The proper choice of the scale dependence $g=g(a)$ makes such theory free of divergences by construction.