scattering amplitudes for $\theta^{\mu\nu}\neq 0$, we need a formula for twisted Wighman functions $W_{N+M}^\theta$ in (\[GNtheta\]) in terms of the untwisted Wightman functions $W_{N+M}^0$. We have already explained this formula: the passage from $W_{N+M}^0$ to $W_{N+M}^\theta$ is achieved by twisting all fields except the gauge fields for unbroken gauge symmetries $H$. Further if $D(V)^{\theta = 0}$ is the connection with gauge potential $V$ for the unbroken group $H$, then the covariant derivative for a twisted matter field $\psi^\theta$ is to be defined as $D_\mu^\theta(V)\psi^\theta :
= (D_\mu(V)^0\psi^0) e^{\frac{1}{2} \overleftarrow{\partial} \wedge
P}$.

This rule preserves the asymptotic conditions and shows that the spectrum of the theory is unaltered by changing $\theta^{\mu\nu}$. It covers theories with spontaneous symmetry breakdown as well provided we have a scheme for treating it for $\theta^{\mu\nu} = 0$. We remark in this connection that since twist factors with time derivatives occur in (\[GNtheta\]) within the time-ordering symbol and the amount of time-translation they generate depend on external momenta, the Gell-Mann-Low formula has to be modified significantly.

Spontaneous Symmetry Breaking
=============================

This topic is of sufficient importance that we discuss it separately. The consistency of this discussion with what we discussed earlier will be apparent.

Let us first 