Abstract
We review and elaborate the complex effective action at one-loop and at zero or finite temperature in the in-out formalism for scalar QED and the vacuum persistence in time-dependent electric fields. Using the gamma-function regularization, we find the effective action in the proper-time integral and the pair-production rate in an exponentially increasing electric field. We apply the quantum invariant theory to the scalar field in time-dependent electric fields and clarify the meaning of multi-pair states. And pair-production rates are derived when the initial state is the Minkowski vacuum or the adiabatic vacuum and are compared with two representations for the Vlasov equation. Finally, the contour integral method gives the pair-production rate in the exponentially increasing electric field.
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