  An axiomatic quantum field theory applied to the self-interacting boson field
is realised in terms of generalised operators that allows us to form products
and take derivatives of the fields in simple and mathematically rigorous ways.
Various spaces are explored for representation of these operators with this
exploration culminating with a Hida-Colombeau algebra. Rigorous well defned
Hamiltonians are written using ordinary products of interacting scalar fields
that are represented as generalised operators on simplified Hida-Colombeau
algebras.
