  We study the path-integral formalism in the imaginary-time to show its
validity in a case with a metastable ground state. The well-known method based
on the bounce solution leads to the imaginary part of the energy even for a
state that is only metastable and has a simple oscillating behavior instead of
decaying. Although this has been argued to be the failure of the Euclidean
formalism, we show that proper account of the global structure of the
path-space leads to a valid expression for the energy spectrum, without the
imaginary part. For this purpose we use the proper valley method to find a new
type of instanton-like configuration, the ``valley instantons''. Although
valley instantons are not the solutions of equation of motion, they have
dominant contribution to the functional integration. A dilute-gas approximation
for the valley instantons is shown to lead to the energy formula. This method
extends the well-known imaginary-time formalism so that it can take into
account the global behavior of the theory.
