7. Conclusion

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 $\tau$. 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 $\hat{H} \left \vert \Psi \right \rangle = 0$ for the physical states. However, the explicit construction of $\hat{H}$ as an hermitian operator acting on a Hilbert space is by no means straightforward. The sign multiplicity of the $\beta$-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.