are the electron mass and charge, respectively, $k_{i}$ is the transverse momentum and $E_0$ is the (constant) electric field. In the present paper we use units so that $\hbar =c =k_B =1$. In these units, $e$ is equal to $\alpha^{1/2}$ and $1$ MeV$=5.064\times 10^{-3}$ fm$^{-1}$.

In most of the physical applications the proper time method introduced by @Sch51 has been used to calculate the pair production rate. The real part of the effective action leads to vacuum polarization and the imaginary part to pair production. Though that method is conceptually well defined and technically rigorous, it is generally difficult to apply it to concrete physical problems such as inhomogeneous electromagnetic fields. Schwinger’s result was generalized to electric fields $E_3=E_3\left(x_{\pm }\right)$, which depend upon either light cone coordinates $x_{\pm }=x_3\pm x_0$, but not upon both, in @To00. The form of the result is exactly the same as in the original formula given by Eq. (\[prod\]). The case of electric fields depending on both $x_+$ and $x_{-}$, $E_3=E_3\left(x_+,x_{-}\right)$ was considered in @Av03.

An alternative approach to pair creation was initiated by @Ca79 [@Ca80], who re-derived Schwinger’s pair production rate by semi-classical tunnelling calculations. The boson and fermion pair production rate by strong 