quantity $w(x)$ can be interpreted as the pair production rate per unit time and unit volume at the space-time point $x=\left(
x_{0},x_{1},x_{2},x_{3}\right) $ [@Gr85; @Ma89].

An alternative description of the pair creation process can be obtained by assuming that the vacuum decays if there exists an ingoing antiparticle mode which is at the same time an outgoing particle mode. An in-vacuum is defined via the ingoing (anti)-particle basis, $_{-}a_{l}\left| {\rm in},{\rm
vac}\right\rangle =_{+}a_{k}\left| {\rm in},{\rm vac}\right\rangle
=0$, whereas the out-vacuum is defined by the outgoing states $^{-}a_{l}\left| {\rm out},{\rm vac}\right\rangle =^{+}a_{k}\left|
{\rm out},{\rm vac}\right\rangle =0$, where $a$ are the particle creation and annihilation operators, respectively [@So82]. The number of the outgoing particles in the mode $k$ created in the in-vacuum is $N_{kk^{\prime }}\equiv \left\langle {\rm
vac},{\rm in}\right| ^{+}a_{k}^{+}$ $^{+}a_{k^{\prime }}\left|
{\rm vac},{\rm in}\right\rangle =\sum_{l}\left| \beta _{kl}\right|
^{2}\delta \left( k-k^{\prime }\right) $, where the coefficients $\beta _{kl}$ are the elements of the single-particle $S$ matrix. Due to the validity of the energy and momentum conservation laws, one can generally choose $\beta _{kl}=\beta _{k}\delta _{kl}$ and thus the number of created particles in the mode $k$ is simply given by $N_{kk^{\prime }}=\left| \beta _{k}\right| ^{2}\delta
\left( k-k^{\prime }\right) $. The delta function 