will refer to it in Sec. \[results\].

Spinor QED {#FF}
----------

The spinor QED formulation of the pair production from vacuum by a time-dependent pulsed electric field has been presented in Ref. [@KTK] (see, also Refs. [@Grib1; @Grib2]). It is based on the Dirac equation that describes spin 1/2 particles, when coupled to an external electromagnetic field. As it was shown there, the dynamics of a single eigenmode of fermionic field, specified by the momentum ${\bm p}$ and the spin projection $\lambda$, is defined by two differential equations, $${\mathrm{i}}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\begin{bmatrix}
                     c_{\bm p}^{(1)}(t)\\
                     c_{\bm p}^{(2)}(t)
                    \end{bmatrix}
                    =
                    \begin{pmatrix} \omega_{\bm p}(t) & {\mathrm{i}}\Omega_{\bm p}(t) \cr -{\mathrm{i}}\Omega_{\bm p}(t) & -\omega_{\bm p}(t) \end{pmatrix}
                    \begin{bmatrix}
                     c_{\bm p}^{(1)}(t)\\
                     c_{\bm p}^{(2)}(t)
                    \end{bmatrix},
\label{a40}$$ that are analogous to Eq. . This similarity is due to the fact that, for a linearly polarized electric field, the bi-spinor part of the fermionic wave function decouples [@KTK; @Grib1; @Grib2]. Hence, each eigenmode exhibits the same time-evolution irrespectively of the particles spins $\lambda$. This does not hold for a circularly or an elliptically polarized fields. In these cases, the Dirac-Heisenberg-Wigner approach [@IBB1] and its development based on the spinoral decomposition [@IBB2] can be used instead. Going back to Eq. , it has to 