Scale-Dependent Functions, Stochastic Quantization and Renormalization

• Main Author: Mikhail V. Altaisky
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2006-04-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: wavelets; quantum field theory; stochastic quantization; renormalization
• Online Access: http://www.emis.de/journals/SIGMA/2006/Paper046/

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 $phi(b)in L^2({mathbb R}^d)$ 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 $phi_a(b,cdot)$ defined on the affine group $G:x'=ax+b$, $a>0,x,bin {mathbb R}^d$, which consists of dilations and translation of Euclidean space. The fields $phi_a(b,cdot)$ 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.