  In quantum cosmology, one often considers tunneling phenomena which may have
occurred in the early universe. Processes requiring quantum penetration of a
potential barrier include black hole pair creation and the decay of vacuum
domain walls. Ideally, one calculates the rates for such processes by finding
an instanton, or Euclidean solution of the field equations, which interpolates
between the initial and final states. In practice, however, it has become
customary to calculate such amplitudes using the No-Boundary Proposal of Hartle
and Hawking. A criticism of this method is that it does not use a single path
which interpolates between the initial and final states, but two disjoint
instantons: One divides the probability to create the final state from nothing
by the probability to create the initial state from nothing and decrees the
answer to be the rate of tunneling from the initial to the final state. Here,
we demonstrate the validity of this approach by constructing continuous paths
connecting the ingoing and outgoing data, which may be viewed as perturbations
of the set of disconnected instantons. They are off-shell, but will still
dominate the path integral as they have action arbitrarily close to the
no-boundary action. In this picture, a virtual domain wall, or wormhole, is
created and annihilated in such a way as to interface between the disjoint
instantons. Decay rates calculated using our construction differ from decay
rates calculated using the No-Boundary Proposal only in the prefactor; the
exponent, which usually dominates the result, remains unchanged.
