Action principle of Bogoliubov coefficients

It is shown that the Schwinger action principle uniquely determines the Bogoliubov coefficients connecting in- and out-states for an oscillator with a time-dependent spring constant, and hence for a charged particle field in a constant homogeneous electric field. This allows a complete constructionn of the S-matrix in the external field approximation. Especially it allows an easy calculation of , the vacuum-to-vacuum amplitude, in this approximation. The method is applied to electromagnetism embedded in a non-abelian gauge group