${\varphi }^4$ Field theory on a Lie group

The ${\varphi }^4$ field model is generalized to the case when the field $\varphi (x)$ is defined on a Lie group: $S[\varphi ]={\int }_{x\in G}L[\varphi (x)]d\mu (x)$, $d\mu (x)$ is the left-invariant measure on a locally compact group $G$. For the particular case of the affine group $G:x^{\mathrm{\prime }}=ax+b,a\in {\mathbb{R}}_{+},x,b\in {\mathbb{R}}^n$ t he Feynman perturbation expansion for the Green functions is shown to have no ultra-violet divergences for certain choice of $\lambda (a)\sim a^{\nu }$.