  We describe in detail a physical situation in which instantons are
necessarily complex, not just Wick rotations of classical solutions to
Euclidean spacetime. These complex instantons arise in the semiclassical
evaluation of vacuum pair production rates, based on Feynman's worldline path
integral formulation. Even though the path integral is a sum over all real
closed trajectories in spacetime, the semiclassical description of
non-perturbative pair production is dominated by closed classical trajectories
that are generically complex. These closed trajectories contain segments
associated with nonperturbative instanton suppression factors as well as
segments producing phase factors that incorporate quantum interference effects.
For a class of time-dependent electric fields we implement this procedure and
demonstrate excellent quantitative agreement with alternative methods.
