$$U(t_1,t_2) = T \exp \left( -i \int_{t_1}^{t_2} dt H_I (t) \right)$$ and $$\phi(\cdot,t) = U(0,t) \phi_F(\cdot,t) U(t,0)
\label{heisenberg}$$

Gell-Mann and Low show that $$G_{N+M}(x_1,\cdots x_{N+M})= \frac{\langle 0,{\rm F}|T \left(\phi_F(x_1) \cdots
  \phi_F(x_{N+M}) e^{i\int d^4 x {\cal L}_I (x)}\right) |0,{\rm
  F}\rangle}{\langle 0,{\rm F}| e^{i\int d^4 x {\cal L}_I (x)}|0,{\rm
  F}\rangle}$$ where $|0,{\rm F}\rangle$ is the vacuum of the free Hamiltonian $H_0$: $$H_0 |0, {\rm F}\rangle = 0 .$$

The proof is standard and will be omitted here.

The twisted quantum fields
--------------------------

Let us look at the case when $\theta^{\mu\nu} \neq 0$, first focussing on the situation with no gauge fields. Gauge fields will be treated later.

Our assumption is that the noncommutative field theory comes from the commutative one by the replacement $$\phi_\theta = \phi_0 e^{\frac{1}{2} \overleftarrow{\partial} \wedge
  P}. 
\label{twistfield}$$ for [*matter*]{} fields, whereas gauge fields are [*not*]{} twisted (See also (\[twistD\])). As $t\rightarrow ~\pm\infty$, the Heisenberg field $\phi_0$ for $\theta_{\mu\nu}~=~0$ becomes the corresponding in- and out- fields $\phi_0^{in,out}$. As for $P_\mu$, they are not affected by these limits, being constants of motion. Hence formally, we find, for the in- and out- fields $\phi^{in,out}_\theta$ of $\phi_\theta$, the results $$\phi_\theta \rightarrow \phi_\theta^{\rm in,out}~:~= \phi_{\rm in, out}
e^{\frac{1}{2}\overleftarrow{\partial} \wedge P}
\quad {\rm as} 