below.

Perturbation Theory for Wightman Functions
------------------------------------------

Perturbation theory for Wightman functions is available [@ostendorf]. We can also construct this formalism directly. For free fields, Wightman functions are Gaussian correlated. For example, $$\langle \phi_F(x_1)\phi_F(x_2)\phi_F(x_3)\phi_F(x_4) \rangle= \langle
\phi_F(x_1)\phi_F(x_2)\rangle \langle\phi_F(x_3)\phi_F(x_4) \rangle
+{\rm permutations}
\label{wightman}$$ while for Dirac fields, there are signs attached to the succesive terms reflecting the signature of the permutations in $x_1,x_2,x_3,x_4$. The two-point functions here are well-known. For example, $$\langle \phi_F(x_1)\phi_F(x_2)\rangle = \Delta_+(x_1~-~x_2,m^2).$$ Now to calculate Wightman functions for interacting fields, we can expand Heisenberg fields $\phi$ in terms of free fields using (\[heisenberg\]) and express the resultant free field correlators in terms of two point functions.

As an illustration, consider $$\begin{aligned}
\lefteqn{\langle T(\phi_F(x_1)\phi_F(x_2)\phi_F(x_3)\phi_F(x_4))
  \rangle =} \nonumber \\
&&\theta(x_1^0-x_2^0)\theta(x_2^0-x_3^0) \theta(x_3^0-x_4^0)
\langle \phi_F(x_1)\phi_F(x_2)\phi_F(x_3)\phi_F(x_4) \rangle + \cdots
\label{wightman1}\end{aligned}$$ The Wightman function is then given by (\[wightman\]). We can similarly calculate the remaining terms in (\[wightman1\]).

Renormalisation theory for Wightman functions has also been developed [@ostendorf].

Summary of our Rules for Twisted Quantum Field Theories
=======================================================

Our rules for transition from the $\theta^{\mu\nu}=0$ to the $\theta^{\mu\nu}\neq 0$ theory are simple and definite. Let us focus on scattering amplitudes. They are given by reduction formulae as in (\[ncLSZ\]). They show that to compute 