being the [*total*]{} four momentum of all fields including interactions.

We will discuss spontaneous symmetry breakdown in the presence of twists later. Here we remark only the following. If the connected, compact Lie group $G$ is the gauged group and it is spontaneously broken to the gauge theory of the subgroup $H$, then the vector fields acquiring mass are to be twisted. Only the gauge fields of $H$ escape the twist.

The LSZ approach to the scattering theory of such interactions appears to be more streamlined and non-perturbative as compared to our earlier treatments[@bpq]. There we used the interaction representation. In this representation, the free Hamiltonian is not twisted whereas the interaction Hamiltonian is $$H_\theta^I = \int d^3 x [{\cal H}_\theta^{M-G} + {\cal H}_\theta^{G}]$$ Here ${\cal H}_\theta^{M-G}$ and ${\cal H}_\theta^{G}$ correspond to the interaction Hamiltonian densities with matter and gauge fields, and with gauge fields alone respectively. The $\theta$- dependence of the interaction representation S-matrix disappears when ${\cal{H}}^G_\theta~=~0$, but that is not the case when both ${\cal{H}}_\theta ^M$ and ${\cal{H}}_\theta^G$ are present. Scattering processes that involve cross terms between ${\cal
H}_\theta^{M-G}$ and ${\cal H}_\theta^{G}$, like the $qg \rightarrow
qg$ scattering in QCD, show effects of noncommutativity.

As we will later 