The next case study is somewhat more subtle and we discuss it to
illustrate the applicability of our effective method. To the extent
that this system may arise as a fluctuation of the gravitational
vacuum, we interpret it as a possible constituent of the Planckian
spacetime foam even though it does not correspond to any class of
wormholes. What we have in mind is the nucleation of a
false vacuum deSitter bubble in a Minkowski background which
represents the key mechanism proposed in Ref. [11], and
expanded in Ref. [10], to generate quantum mechanically an
inflationary domain. From our
vantage point, the difficulty of that proposal is the presence of
a virtual black hole, decaying through Hawking radiation,
as an intermediate state between the initial Minkowski
state and the final MinkowskideSitter state. This intermediate
state involves the still obscure issue of the final stage
of black hole evaporation. Interestingly enough, it is possible to
bypass this difficulty by choosing a
negative surface tension and vanishing total mass energy
for the original quantum fluctuation that triggers the process in
the first place. Physically, this assumption of negative surface
tension may be justified by a simple analogy with the multiphase
vacuum of QCD. If the unified field theory undergoing primordial
phase transitions is of the Ginzburg-Landau type, then, for some
choice of the coupling constants, bags can form around test charges
with positive volume and negative surface energy
[16]. The sign of the surface tension follows from the
negative condensation energy. The vacuum state for such a model
behaves as a type II superconductor with maximal
boundary surface between the normal (non-confining) phase and
the ordered (confining) phase [17].
With the above choice of surface tension and vanishing total mass-energy, the initial and final states are degenerate in energy and a spontaneous transition between them is allowed without an intermediate blackhole state.
Some preliminary remarks on the classical
dynamics of a deSitter bubble will be helpful in order to clarify
our final result.
The classical equation of motion for the bubble trajectory is
Having assumed a negative surface tension, we retrace our steps as
in the previous case study: the matching condition now is
Presently, for reasons of clarity and conciseness, we choose to
discuss the case
.
The corresponding
semi-classical solution, which describes the nucleation of an
expanding deSitter bubble, is obtained by matching the expanding
half of the classical bounce to a quantum tunneling solution. Then, the
classical turning point acquires the meaning of nucleation radius. The
corresponding lagrangian and hamiltonian are,
Our last comment concern the physical interpretation of this result.
The model discussed above describes the quantum birth of an
inflationary bubble in a Minkowski background. In connection with
this process,
we find that there is a lingering ambiguity in the published
literature. It is indeed interesting, and perhaps somewhat puzzling,
that the initial radius and the nucleation rate in this case are the
same as for the false vacuum decay, that is, the nucleation of a
Minkowski bubble in a deSitter background, originally discussed by
Coleman and DeLuccia [18]. With hindsight, this
coincidence is hardly surprising since the two cases
appear to be completely symmetrical due to the fact that the
euclidean trajectory interpolating between the two vacuum states is
the same in both cases. However, there is a difference, even
at the classical level, and it
lies in the global structure of the spacetime manifold in the two
cases. The point is that, for an inflationary bubble in a Minkowski
background, at a given instant, say the nucleation Minkowski time,
all the points in the interval
suddenly undergo a
phase transition
from the Minkowski to the deSitter geometry. Then, the new vacuum
domain, driven by the negative pressure of the false vacuum,
expands exponentially, eventually filling up the whole spacetime.
In contrast, a true vacuum bubble, no matter how large,
will never fill up the whole deSitter manifold. As a matter of
fact, in this difference lies the problem of the ``graceful exit''
from the inflationary stage [19].
Stefano Ansoldi
Department of Theoretical Physics
University of Trieste
TRIESTE - ITALY