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 $\{{\varphi }_a(x)\}$ 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 ${\mathrm{\Gamma }}_{(A)}$ 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—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 renormalization group results for ${\varphi }^4$ model are reproduced.

Figure 1

Renormalized vertex functions. The renormalized inverse propagator and the renormalized vertex functions in ${\varphi }^4$ theory are shown in a one-loop approximation. Here and after, we assume $-\lambda$ value for each vertex.