In this paper we have formulated
the classical and quantum dynamics of a spherically
symmetric shell in a de Sitter-Schwarzschild background, directly in
terms of the Einstein-Hilbert action supplemented by an arbitrary
equation of state. An effective action for the shell radial degree
of freedom was then obtained following the FGG-reduction technique.
A distinctive feature of our formulation is its invariance under a
general redefinition of the evolution parameter. This feature leads
to the Hamiltonian constraint H=0. This constraint is discussed further in
Appendix A, where we have compared our variational procedure
with the FGG-approach, in order to check the consistency of our
result. In our formulation, however, the vanishing of H, far from
being a fortuitous coincidence which makes the true classical dynamics
and the naive classical dynamics identical [19],
acquires a precise mathematical and physical significance, namely, that
reparametrization invariance implies that there
is no energy associated with the evolution parameter .
This, in a nutshell, is the simple message on which our approach is
based: once unfolded and reinterpreted by the machinery of the canonical
formalism, that message translates into the matching
condition(1), and effectively controls the classical
evolution of the system which was briefly reviewed in Section IV. At
the quantum level, however, any two
approaches based on a different choice of evolution parameter may
differ significantly. Thus, building on our classical result,
we have laid the foundations of our
quantum approach and explored some of its consequences.
We have shown that the vanishing of the Hamiltonian, in a weak
sense, can now be interpreted as the Wheeler-De Witt equation
for the
physical states. However, the explicit construction of
as
an hermitian operator acting
on a Hilbert space is by no means straightforward. The sign
multiplicity of the
-functions,
non-locality and ordering ambiguities are the major
limitations to the full utilization of our formulation. However, to
the extent that our approach is analogous to the minisuperspace
approach to quantum cosmology, it seems to us that the above
difficulties may represent a shadow of deeper problems which are
widely suspected to be a general feature of quantum gravity [15].
At any rate, because of the above difficulties, we limited our
considerations to the quantum dynamics of a shell in the WKB approximation. In
particular, we have studied the probability of quantum tunneling
under the classical potential barrier, and have shown that vacuum
decay can be described by such a tunneling process.
All known results on decay probabilities and nucleation
radii are correctly reproduced by our formalism.
In addition, we have speculated on the nature of some rather exotic
processes which we have interpreted as ``creation of vacuum domains
from nothing" . However, a proper treatement of such processes lies
beyond our first quantized formulation of shell dynamics. One
possible step in this direction would be to apply the Dirac
formalism of canonical quantization not only to the shell, but to
the gravitational field as well[14,15],
[20]. In such a case, the whole spacetime
enters the theory as a geometrodynamical entity.
After this paper was submitted for publication, a similar formulation
was proposed in [21] to describe massive dust shells.
Stefano Ansoldi
Department of Theoretical Physics
University of Trieste
TRIESTE - ITALY