$\phi^4$ Field theory on a Lie group

TL;DR

This paper extends the $^4$ field theory to fields defined on Lie groups, demonstrating divergence-free perturbation expansions for specific group-dependent coupling functions.

Contribution

It introduces a generalized $^4$ model on Lie groups and analyzes its perturbation expansion, revealing conditions for divergence-free behavior.

Findings

Perturbation expansion has no ultra-violet divergences for certain coupling functions.

The model is explicitly constructed on the affine group.

Conditions on the coupling function ^ determine divergence properties.

Abstract

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