  An interesting idea is that the universe could be spontaneously created from
nothing, but no rigorous proof has been given. In this paper, we present such a
proof based on the analytic solutions of the Wheeler-DeWitt equation (WDWE).
Explicit solutions of the WDWE for the special operator ordering factor p=-2
(or 4) show that, once a small true vacuum bubble is created by quantum
fluctuations of the metastable false vacuum, it can expand exponentially no
matter whether the bubble is closed, flat or open. The exponential expansion
will end when the bubble becomes large and thus the early universe appears.
With the de Broglie-Bohm quantum trajectory theory, we show explicitly that it
is the quantum potential that plays the role of the cosmological constant and
provides the power for the exponential expansion of the true vacuum bubble. So
it is clear that the birth of the early universe completely depends on the
quantum nature of the theory.
