as ingoing, relabel them as $q_1,q_2,\cdots q_{N+M}$ and write $$\langle -q_1,q_2, \cdots -q_N; {\rm out} \mid q_{N+1},\cdots q_{N+M}; {\rm in}\rangle~=~\int {\cal{I}}~G_{N+M},$$ where $${\cal{I}}~=~\prod^{N+M}_{i=1} d^4x_i~e^{-iq_i\cdot x_i}~i(\partial_i^2+m^2)
\label{II}$$ and $$G_{N+M} (x_1,\cdots,x_{N+M})~=~\langle 0\mid T(\phi(x_1)\cdots \phi(x_{N+M})\mid 0 \rangle.
\label{LSZformula}$$

We will later see the differences in the scattering amplitude on the GM plane through an analysis of the Gell-Mann-Low formula.

The Gell-Mann-Low Formula
-------------------------

The Heisenberg fields $\phi$ and the free fields $\phi_F$ at time $t~=~0$ fulfill the [*same*]{} canonical algebra if the interaction has no time derivatives. We assume that to be the case.

Then in perturbation theory, we choose the same representation of the canonical algebra at time 0, namely that of the free field $\phi_F$: $$\phi(\cdot, 0) = \phi_F(\cdot,0) .$$ This implies that $$\begin{aligned}
\phi_F(\cdot, t) &=& e^{iH_0 t}\phi_F(\cdot,0)e^{-i H_0 t}, \\
\phi(\cdot,t) &=& e^{i H t}\phi(\cdot,0)e^{-i H t} = e^{i H
  t}\phi_F(\cdot,0)e^{-i H t} \end{aligned}$$ or $$\phi(\cdot,t) = (e^{i H t}e^{-i H_0 t} \phi_F(\cdot,t)(e^{i H_0 t}
e^{-i H t})$$ Let us define $$U(t_1,t_2) = e^{i t_1 H_0}e^{-i(t_1-t_2)H}e^{-it_2 H_0}$$ Then $$\begin{aligned}
U(t,t) &=& 1, \\
i \frac{\partial U}{\partial t_1}(t_1,t_2) &=& H_I(t_1) U(t_1,t_2),\end{aligned}$$ where $$H_I(t) = e^{it_1 H_0} H_I(0) e^{-i t_1 H_0}, \quad H_I(0) = H_I$$ is the interaction representation Hamiltonian.

Thus 