Scale-dependent functions, stochastic quantization and renormalization
This paper studies the possibility to unify some methods of regularization, such as the renormalization group method, stochastic quantization etc., by means of an extension of the standard field theory of the square-integrable functions \(\phi(b) \in L^2(R^d)\) to the theory of functions depending on the coordinate \(b\) and ``resolution'' parameter \(a\). In the simplest case such field theory turns out to be a theory of fields \(\phi_a(b,\cdot)\) defined on the affine group \(G\) of the maps \(x\mapsto x' = ax+b\), \(a > 0\), \(x, b \in R^d\), which consists of dilations and translation of Euclidean space. The fields \(\phi_a(b,\cdot)\) are obtained making use of the continuous wavelet transform. The parameters of the theory may explicitly depend on the resolution \(a\). A suitable choice of the scale dependence \(g = g(a)\) makes such theory free of divergences by construction.
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