$ϕ^4$ Field theory on a Lie group

Abstract

The $ϕ^4$ field model is generalized to the case when the field $ϕ(x)$ is defined on a Lie group: $S[ϕ]=\int_{x\in G} L[ϕ(x)] dμ(x)$, $dμ(x)$ is the left-invariant measure on a locally compact group $G$. For the particular case of the affine group $G:x'=ax+b,a\in\R_+, x,b \in \R^n$ t he Feynman perturbation expansion for the Green functions is shown to have no ultra-violet divergences for certain choice of $λ(a) \sim a^ν$.