0 Field theory on a Lie group

The φ 4 field model is generalized to the case when the field φ(x) is defined on a Lie group: S[φ] = x∈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 ∈ R+, x, b ∈ R n the Feynman perturbation expansion for the Green functions is shown to have no ultraviolet divergences for certain choice of λ(a) ∼ a ν. The scalar field theory with the forth power interaction λ 4! φ 4 (x) defined on Euclidean space x ∈ R n is one of the most instructive models any textbook in field theory starts with, see e.g. [1]. Often called a Ginsburg-Landau model for its ferromagnetic applications, the model describes a quantum field with the Lagrangian L[φ] = 1 2 (∂ µ φ) 2 + m 2