Scale-Dependent Functions, Stochastic Quantization and Renormalization

Details Scale-Dependent Functions, Stochastic Quantization and Renormalization Scale-Dependent Functions, Stochastic Quantization and Renormalization

2006-04-24

arxiv.org/abs/hep-th/0604170v1

SIGMA 2:046,2006

Physics

High Energy Physics

High Energy Physics - Theory

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

Scientific paper

10.3842/SIGMA.2006.046

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 $G:x'=ax+b$, $a>0,x,b\in {\mathbb R}^d$, 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.

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