\cdot q_J - \sum_{J<I-1} \theta\cdot q_J\right), \\
x_I^0 &\rightarrow& x_I^0 - \frac{1}{2} \left(\sum_{J>I}
  \theta \cdot q_J - \sum_{J<I} \theta\cdot q_J\right), \\
x_{I+1}^0 &\rightarrow& x_{I+1}^0 - \frac{1}{2} \left(\sum_{J>I+1}
  \theta \cdot q_J - \sum_{J<I+1} \theta\cdot q_J\right)\end{aligned}$$ Or $$x_{I-1}^0 - x_I^0 \rightarrow (x_{I-1}^0 - x_I^0) - \frac{1}{2} \theta
\cdot q_J + \frac{1}{2} \sum_{J \neq I-1,I} \theta \cdot q_J.$$ But $\sum \vec{q}_J = 0$, so that $$x_{I-1}^0 - x_I^0 \rightarrow x_{I-1}^0 - x_I^0~ - ~\frac{1}{2} \theta
\cdot q_{I-1}~-~  \frac{1}{2} \theta \cdot q_I
\label{diff1}$$ Similarly, $$x_I^0 - x_{I+1}^0 \rightarrow x_I^0 - x_{I+1}^0 ~-~ \frac{1}{2} \theta
\cdot q_I ~-~ \frac{1}{2} \theta \cdot q_{I+1}
\label{diff2}$$ From (\[diff1\],\[diff2\]), we see that each $x_I^0$ is (time) shifted to $$x_I^0 + \delta x_I^0, \quad \delta x_I^0 = \delta x_I^0 (q_1, \cdots q_N).$$ where the $\delta x_I^0$ actually depend on the ordering on $x_I^0$. No further simplification seems possible.

We emphasize the following important observations.

-   Firstly, (\[ncLSZ\]) involves only the $\theta^{\mu\nu}=0$ fields in $W_N^0$. So it can be used to map any commutative theory to noncommutative one, including also the standard model. But special care is needed to treat gauge fields. Gauge fields are [*[not]{}*]{} twisted unlike matter fields. As explained elselwhere, this means that the Yang-Mills tensor is not 