Wavelet regularization of gauge theories

TL;DR

This paper introduces a novel gauge theory framework using wavelet regularization on Lie groups, ensuring finiteness by construction and relating charges across scales, with specific focus on SU(N) gauge groups.

Contribution

It extends gauge invariance principles to fields on Lie groups using wavelet methods, creating a finite, scale-relating gauge theory framework.

Findings

The constructed theory is finite by design.

The renormalization group relates charges at different scales.

Application to SU(N) gauge groups demonstrates the approach.

Abstract

Extending the principle of local gauge invariance $\psi(x)\to \exp\left(\imath \sum_A \omega^A(x)T^A \right) \psi(x), x \in \mathbb{R}^d$, with $T^A$ being the generators of the gauge group $\mathcal{A}$, to the fields $\psi(g)\equiv \langle \chi|\Omega^*(g)|\psi\rangle$, defined on a locally compact Lie group $G$, $g\in G$, where $\Omega(g)$ 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.