Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 046, 17 pages      hep-th/0604170      https://doi.org/10.3842/SIGMA.2006.046

Scale-Dependent Functions, Stochastic Quantization and Renormalization

Mikhail V. Altaisky a, b
a) Joint Institute for Nuclear Research, Dubna, 141980 Russia
b) Space Research Institute RAS, Profsoyuznaya 84/32, Moscow, 117997 Russia

Received November 25, 2005, in final form April 07, 2006; Published online April 24, 2006

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) Î 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 Î 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.

Key words: wavelets; quantum field theory; stochastic quantization; renormalization.

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