Unifying renormalization group and the continuous wavelet transform

It is shown that the renormalization group turns to be a symmetry group in a theory initially formulated in a space of scale-dependent functions, i.e., those depending on both the position $x$ and the resolution $a$. Such a theory, earlier described in [1,2], is finite by construction. The space of scale-dependent functions ${{\ensuremath{\phi}}_{a}(x)}$ is more relevant to a physical reality than the space of square-integrable functions ${\mathrm{L}}^{2}({\mathbb{R}}^{d})$; because of the Heisenberg uncertainty principle, what is really measured in any experiment is always defined in a region rather than a point. The effective action ${\mathrm{\ensuremath{\Gamma}}}_{(A)}$ of our theory turns out to be complementary to the exact renormalization group effective action. The role of the regulator is played by the basic wavelet\char22{}an ''aperture function'' of a measuring device used to produce the snapshot of a field $\ensuremath{\phi}$ at the point $x$ with the resolution $a$. The standard renormalization group results for ${\ensuremath{\phi}}^{4}$ model are reproduced.