  The in-out formalism is a systematic and powerful method for finding the
effective actions in an electromagnetic field and a curved spacetime provided
that the field equation has explicitly known solutions. The effective action
becomes complex when pairs of charged particles are produced due to an electric
field and curved spacetime. This may lead to a conjecture of one-to-one
correspondence between the vacuum polarization (real part) and the vacuum
persistence (imaginary part). We illustrate the one-loop effective action in a
constant electric field in a Minkowski spacetime and in a uniform electric
field in a two-dimensional (anti-) de sitter space.
