Boundary completion of vacuum persistence probability

The vacuum persistence probability, encoded in the imaginary part of the in-out effective action, is a basic measure of vacuum instability and particle production. However, its evaluation may appear prescription-dependent: the Bogoliubov prescription and the Green's function prescription can yield different expressions. We show that this apparent ambiguity arises because the Green's function prescription omits the nontrivial contribution from the endpoint vacuum wavefunctionals, which should be present in the complete in-out amplitude. Including this contribution accomplishes the boundary completion of the Green's function prescription. The ambiguity is thereby resolved, and the complete result agrees with the Bogoliubov expression.