Wavelet regularization of gauge theories

Abstract

Extending the principle of local gauge invariance $ψ(x)\to \exp\left(\imath \sum_A ω^A(x)T^A \right) ψ(x), x \in \mathbb{R}^d$, with $T^A$ being the generators of the gauge group $\mathcal{A}$, to the fields $ψ(g)\equiv \langle χ|Ω^*(g)|ψ\rangle$, defined on a locally compact Lie group $G$, $g\in G$, where $Ω(g)$ is suitable square-integrable representation of $G$, it is shown that taking the coordinates ($g$) on the affine group, we get a gauge theory that is finite by construction. The renormalization group in the constructed theory relates to each other the charges measured at different scales. The case of the $\mathcal{A}=SU(N)$ gauge group is considered.