be solved with the same initial conditions as in Sec. \[BB\], $c_{\bm p}^{(1)}(t_0)=1$ and $c_{\bm p}^{(2)}(t_0)=0$. This time, however, $$\displaystyle\Omega_{\bm p}(t)=-\frac{ce{\cal E}(t)\epsilon_\perp}{2\omega_{\bm p}^2(t)},
\label{wrzuta5}$$ where $\epsilon_\perp=\sqrt{(c{\bm p}_\perp)^2+(mc^2)^2}$. Another difference is that while the matrix governing the time evolution here is Hermitian, for the bosonic case it is pseudo-Hermitian \[see, Eq. \]. This, in principle, may have far-reaching consequences which will be studied in detail next.

In closing this section, we note that the charge distribution of created fermion pairs with momenta ${\bm p}$ and $-{\bm p}$ for a particle and an anti-particle, respectively, is $$Q^{(1/2)}({\bm p},t)=e|c_{\bm p}^{(2)}(t)|^2,\label{a28}$$ where the coefficient $c_{\bm p}^{(2)}(t)$ satisfies .

Momentum distributions of created particles {#results}
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In this section, we shall analyze statistical effects in the electron-positron pair creation from vacuum under the influence of time-dependent, linearly polarized electric field pulses. For this purpose, we will use the formulations introduced in Secs. \[BB\] and \[FF\], treating the pairs as scalar or spinor particles.

We define the momentum distribution of created particles by a sequence of $N_{\rm rep}$ identical electric field pulses, $${\cal P}_{N_{\rm rep}}^{(s)}({\bm p})=\lim_{t\rightarrow +\infty} |c_{\bm p}^{(2)}(t)|^2,
\label{res1}$$ which follows from Eqs.  and . Depending on the statistics, which is reflected in a different 