[@Di_Piazza_2007][^4]): $$\label{D}
\begin{split}
D(A)=&-i\int_0^{\infty}ds\, e^{is(\hat{\Pi}^2-m^2+i\epsilon)}=-i\int_0^{\infty}ds\, e^{-i(m^2-i\epsilon)s}\\
&\times \bigg\{1+\frac{e\hat{n}}{2P_X}
[\hat{A}(\phi+2sP_X)-\hat{A}(\phi)]\bigg\}\\
&\times e^{-i\int_0^sds'\left[{\bm P}_\perp-e{\bm
A}(\phi+2s'P_X)\right]^2}e^{-2isP_\phi P_X},
\end{split}$$ where we have introduced the operators $P_{\phi}=(P_t+P_{x_{\parallel}})/2$ and $P_X=-(P_t-P_{x_{\parallel}})=-(nP)$ of the conjugated momenta to the coordinates $\phi=t-x_{\parallel}$ and $X=(t+x_{\parallel})/2$, with $x_{\parallel}=\bm{n}\cdot\bm{x}$, such that $\phi$ and $X$ can be interpreted as a “time” and a “space” coordinate, respectively, i.e., $[\phi,P_{\phi}]=-i$ and $[X,P_{X}]=i$. Note that $t=X+\phi/2$, $x_{\parallel}=X-\phi/2$, $P_t=P_{\phi}-P_X/2$, and $P_{x_{\parallel}}=P_{\phi}+P_X/2$.

Out of the different partial amplitudes which arise from the above substitutions, we work out only the following one $$\label{M_op_1}
\begin{split}
M^{(1)}=&-e^N\int d^4x\,\text{Tr}\,\langle x|D(A)\text{e}^{-i(k_1x)}[2(\Pi e_1)+\hat{k}_1\hat{e}_1]\cdots \\
&\times D(A)\text{e}^{-i(k_Nx)}[2(\Pi e_N)+\hat{k}_N\hat{e}_N]|x\rangle,
\end{split}$$ which arises from the substitution in 1.. This partial amplitude is always present, independently of the number of the external photons and, as it will also be clear from the considerations below, the analysis of the other partial amplitudes proceeds analogously. By looking at the expression of the operators $D(A)$ (see Eq. (\[D\])), the coordinate operators $X$ and $\bm{x}_{\perp}$ appear to occur only in the exponentials relative to the external photons. By employing the operator identity $e^{i(k_jx)}f(P)e^{-i(k_jx)}=f(P+k_j)$, we can move all the operators $e^{i(k_{j,X}X+\bm{k}_{j,\perp}\cdot\bm{x}_{\perp})}$ to the left and let them act on the bra $\langle x|$. The result is $$\label{M_op_2}
\begin{split}
M^{(1)}=&-e^N\int d^4x\,e^{i(K_XX+\bm{K}_{\perp}\cdot\bm{x}_{\perp})}\text{Tr}\,\langle x|e^{-ik_{1,\phi}\phi}\{2[(\Pi^{\mu}+\kappa^{\mu}_2) e_{1,\mu}]+\hat{k}_1\hat{e}_1\}D_2(A)\\
&\cdots\times e^{-ik_{N-1,\phi}\phi}\{2[(\Pi^{\mu}+\kappa^{\mu}_N) 