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) ∊ L²(Rd) 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 φa(b,·) defined on the affine group G: x′ = ax+b, a > 0, x, b ∊ Rd, which consists of dilations and translation of Euclidean space. The fields φa(b,·) 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