states. Regarding the second diagram, an important issue here arises. This diagam represents a vacuum fluctuation term, which is divergent. As well known [@GreinerFIELD], all vacuum fluctuations diagrams of any order sum up to a phase factor $\left\langle \Omega\left|U_{I}\left(t,0\right)\right|\Omega\right\rangle$, and all divergences cancel each other. Therefore we can write: $$\left\langle \mathbf{p}_{f},s_{f}\left|U_{I}\left(t,0\right)\right|\mathbf{p}_{i},s_{i}\right\rangle =\left\langle \Omega\left|U_{I}\left(t,0\right)\right|\Omega\right\rangle \cdot \left\langle \mathbf{p}_{f},s_{f}\left|U_{I}\left(t,0\right)\right|\mathbf{p}_{i},s_{i}\right\rangle _{\textrm{ext}},$$ where $\left\langle \mathbf{p}_{f},s_{f}\left|U_{I}\left(t,0\right)\right|\mathbf{p}_{i},s_{i}\right\rangle_{\textrm{ext}}$ denotes the contribution from diagrams with external fermionic legs. This vacuum fluctuation term is important because, together with those of Eq. (\[eq:cfhgfd\]), it gives a global phase $\prod_{k=1}^{n}\left\langle \Omega_{k}\left|U_{kI}\left(t,0\right)\right|\Omega_{k}\right\rangle$ independent of $j$, which factorizes out of the sum. Therefore, such terms are physically unimportant, the only relevant part being $\left\langle \mathbf{p}_{f},s_{f}\left|U_{I}\left(t,0\right)\right|\mathbf{p}_{i},s_{i}\right\rangle_{\textrm{ext}}$. From now on, we will work only with diagrams with external fermionic legs, and we drop the pedex “ext”.

Coming back to Eq. (\[eq:dsfdfbv\]), we can now focus our attention on the third and the fourth term, that correspond to the last two diagrams. Since $:\psi_{2b}\bar{\psi}_{1a}:\; = - :\bar{\psi}_{1a}\psi_{2b}:$ for fermions, these two terms give the same contribution. The Dirac propagator reads: $$\begin{aligned}
iS_{ab}\left(x_{1}-x_{2}\right)
& =& 
\sum_{s=1}^{2}\sum_{\mathbf{p}=-\infty}^{+\infty}\frac{1}{L^{3}}\frac{mc^{2}}{E_{p}}\left\{ \theta\left(t_{1}-t_{2}\right)e^{-\frac{i}{\hbar}p^{\mu}\left(x_{1\mu}-x_{2\mu}\right)}u_{a}\left(p,s\right)\overline{u}_{b}\left(p,s\right) \right. \nonumber \\
& & \left.\qquad\qquad\qquad\quad\;\; -\theta\left(t_{2}-t_{1}\right)e^{-\frac{i}{\hbar}p^{\mu}\left(x_{2\mu}-x_{1\mu}\right)}\overline{v}_{b}\left(p,s\right)v_{a}\left(p,s\right)\right\}, \end{aligned}$$ while the matrix element gives: $$\langle \mathbf{p}_{f},s_{f}|:\bar{\psi}_{1a}\psi_{2b}:|\mathbf{p}_{i},s_{i}\rangle 
=
\frac{1}{L^{3}}\frac{mc^{2}}{\sqrt{E_{f}E_{i}}}e^{\frac{i}{\hbar}p_{f}^{\mu}x_{1\mu}}e^{-\frac{i}{\hbar}p_{i}^{\mu}x_{2\mu}}\overline{u}_{a}\left(p_{f},s_{f}\right)u_{b}\left(p_{i},s_{i}\right).$$ In the following we 