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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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