$$_\theta \langle -q_N,-q_{N-1},\cdots -q_1; {\rm out}|q_{N+M},q_{N+M-1},\cdots ,q_{N+1};{\rm
  in}\rangle _\theta ~=~\int {\cal{I}} G_{N+M}^\theta(x_1,x_2,\cdots,x_{M+N})
\label{ncLSZ}$$ where ${\cal{I}}$ is as defined in (\[II\]).

On Fourier transforming as in (\[LSZformula\]), the $\theta_{ij}$ (space-space) part of the twist can be partially integrated. It gives the usual phase $e^{\frac{i}{2} q_I^i \theta_{ij} q_J^j}$. The time step-function (in the time-ordering T) in (\[GNtheta\]) does not affect this manipulation.

To handle the $\theta_{0i}$ part, consider a typical term $$\begin{aligned}
g_{N+M}^\theta (x_1 \cdots x_{N+M}) &=&\theta(x_1^0 - x_2^0) \theta(x_2^0 -
x_3^0) \cdots \theta(x_{N+M-1}^0 - x_{N+M}^0) \nonumber \\
&& e^{\frac{i}{2} \sum_{I<J}
  \partial_{x_I} \wedge \partial_{x_J}} W_{N+M}^0 (x_1, \cdots x_{N+M}).
\label{ncLSZ1}\end{aligned}$$ which occurs on expanding the time-ordered product in terms of retarded products.

The twist here is the product of terms $$e^{\frac{i}{2} [\partial_{x^0_I} \theta \cdot \nabla_J - (\theta \cdot
    \nabla_I) \partial_{x^0_J}]}, \quad I<J, \quad \theta \cdot
    \nabla_J \equiv \theta_{0i} \partial_{x_J^i}
\label{twist}$$ on retaining just $\theta_{0i}$. The coefficient of $\partial_{x^0_I}$ in the exponential is thus $$\sum_{J>I} \theta \cdot \nabla_J - \sum_{J<I} \theta \cdot \nabla_J$$ On partial integration in eq(\[ncLSZ\]) $\nabla_J$ becomes $iq_J$ and $$e^{\frac{i}{2} \partial_{x_I^0} (\sum_{J>I} - \sum_{J<I})\theta \cdot
  \nabla_J} \rightarrow e^{-\frac{1}{2} (\sum_{J>I} \theta \cdot q_J -
  \sum_{J<I} \theta\cdot q_J) \partial_{x_I^0}}$$ This translates the $x_I^0$’s according to $$\begin{aligned}
x_{I-1}^0 &\rightarrow& x_{I-1}^0 - \frac{1}{2} \left(\sum_{J>I-1}
  \theta 