  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.
