\quad t \rightarrow \mp \infty.
\label{phiin}$$ For the twisted in and out annihilation and creation operators $a_k^{in,out},a_k^{\dagger in,out}$ we thus find $$a_k^{in,out}~=~c_k^{in,out}e^{-\frac{i}{2}k_\mu\theta^{\mu\nu}P_\nu},\qquad
a_k^{\dagger in,out}~=~c_k^{\dagger in,out}e^{\frac{i}{2}k_\mu\theta^{\mu\nu}P_\nu}.\label{inout}$$

There is a further convention we want to explain. For consistency with our notation for the coproduct on the Poincaré group[@07081379], we define $$a_k^{\dagger in}|k_r,k_{r-1},\cdots ,k_1\rangle_{\rm in}~=~ |k_r,k_{r-1},\cdots ,k_1,k\rangle_{\rm in}$$ and similarly for the action of $a_k^{\dagger out}$. Thus for example $$|k_r,k_{r-1},\cdots ,k_1\rangle_{\rm in}~=~a_{k_1}^{\dagger in}a_{k_2}^{\dagger in}\cdots 
a_{k_r}^{\dagger in}|0\rangle_{\rm in} .$$ and $$\langle -q_N,-q_{N-1},\cdots -q_1;out|q_{N+M},q_{N+M-1},\cdots ,q_{N+1}\rangle
~=~\int {\cal{I}}~G_{N+M}^\theta(x_1,x_2,\cdots , x_{N+M})
\label{scatter}$$ $G_{N+M}~\equiv G_{N+M}^0$ has got changed to $G_{N+M}^\theta $ for $\theta_{\mu\nu}\neq 0$ where $$\begin{aligned}
G_{N+M}^\theta (x_1, \cdots x_{N+M}) &=& T e^{\frac{i}{2}\sum_{I<J}\partial_{x_I}
\wedge \partial_{x_J}} W_{N+M}^0 (x_1, \cdots x_{N+M}) \nonumber \\
:&=&~T~W_{N+M}^\theta (x_1, \cdots x_{N+M})
\label{GNtheta}\end{aligned}$$ and $W_{N+M}^0$ are the standard Wightman functions for untwisted fields: $$W_{N+M}^0 (x_1, \cdots x_{N+M}) = \langle 0 | \phi_0 (x_1) \cdots \phi_0(x_{N+M})
|0\rangle.$$ It is important that because of translational invariance, the $W_{N+M}^\theta$ (and hence the $G_{N+M}^\theta$) depend only on coordinate differences.

For simplicity, we have included only matter fields, and that too of one type, in (\[scatter\]). Gauge fields can be included too, but they are not acted on by the twist exponential in (\[GNtheta\]).

The scattering matrix element is thus: 