Scale-Dependent Functions, Stochastic Quantization and Renormalization
M. V. Altaĭsky · Symmetry Integrability and Geometry Methods and Applications · 2006
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) ∈ L 2 (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, •) defined on the affine group G : x = ax + b, a > 0, x, b ∈ R d , which consists of dilations and translation of Euclidean space.The fields φ a (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.