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 theory, earlier described in {\em Phys.Rev.D} 81(2010)125003, 88(2013)025015, is finite by construction. The space of scale-dependent functions $\{{\varphi }_a(x)\}$ is more relevant to physical reality than the space of square-integrable functions ${\mathrm{L}}^2(R^d)$, because, due to the Heisenberg uncertainty principle, what is really measured in any experiment is always defined in a region rather than point. The effective action ${\mathrm{\Gamma }}_{(A)}$ of our theory turns to be complementary to the exact renormalization group effective action. The role of the regulator is played by the basic wavelet – an “aperture function” of a measuring device used to produce the snapshot of a field $\varphi$ at the point $x$ with the resolution $a$. The standard RG results for ${\varphi }^4$ model are reproduced.