Department of Radiation and Radiobiological Research,
Joint Institute for Nuclear Research,
141980 Dubna,
E-mail: altaisky@mx.iki.rssi.ru

Scale-dependent functions and stochastic quantization

We consider quantum and random fields on the affine group $G:x'=ax+b$, $a>0$, $x,b\in {\mathbb R}^d$, consisting of dilations and translation of Euclidean space. The fields $\phi_a(b,\cdot)$ are constructed using the continuous wavelet transform. The explicit dependence of the correlation properties of the fields $\phi$ on the scale $a$ is shown to be useful for  the construction of divergence-free field theory. An examples of the scalar field theory and stochastic quantization of
gauge theories are presented.