  Dynamical Sauter-Schwinger mechanism of electron-positron pair creation by a
time-dependent electric field pulses is considered using the $S$-matrix
approach and reduction formulas. They lead to the development of framework
based on the solutions of the Dirac equation with the Feynman- or anti-Feynman
boundary conditions. Their asymptotic properties are linked to the
spin-resolved probability amplitudes of created pairs. The same concerns the
helicity-resolved amplitudes. Most importantly, the aforementioned spin- or
helicity-resolved amplitudes, when summed over spin or helicity configurations,
reproduce the momentum distributions of created particles calculated with other
methods that are typically used in this context. This does validate the current
approach. It also allows us to investigate the vortex structures in momentum
distributions of produced particles, as the method provides an access to the
phase of the probability amplitude. As we also illustrate numerically, the
method is applicable to arbitrary time-dependent electric fields with, in
general, elliptical polarization. This proves its great flexibility.
