the vector potential, $A_S=m_{\mathrm{e}}c/|e|$, as well as the Compton time $t_C=\hbar/m_{\mathrm{e}}c^2$. Here, $m_{\rm e}$ is the electron mass and $e=-|e|<0$ is its charge. Since now on, in our theoretical formulation we shall keep $\hbar=1$ and an arbitrary mass of created particles $m$. However, we will choose $m=m_{\rm e}=c=\hbar=1$ in our numerical calculations.

Theoretical formulation {#theory}
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The spontaneous formation of particle-antiparticle pairs by a homogeneous in space, time-dependent electric field is studied in this paper within the scalar and the spinor QED frameworks. In order to elucidate the differences between both approaches, we present below both theoretical formulations. Typically, such comparison has been performed within the quantum kinetic approach. Thus, concealing very subtle but fundamental features of quantum dynamics. Here, we extend our previous studies [@KTK; @KKproc] to scalar QED. As we show, while the spinor QED facilitates a typical unitary time-evolution of the respective fermionic field eigenmodes [@KTK; @KKproc], the respective time evolution of the bosonic field eigenmodes is pseudo-unitary.

Electric field description {#electric}
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![Time-dependence of the electric field $\mathcal{E}(t)$ and the vector potential $A(t)$ (in relativistic units) for a single pulse considered in this paper ($N_{\rm rep}=1$). The shape functions in  are for $\sigma=5/t_C$, $T_0=40t_C$ (left column), 