1 Introduction

The dynamics of relativistic thin shells is a recurrent topic in the literature about the classical theory of gravitating systems and the still ongoing attempts to obtain a coherent description of their quantum behaviour. Certainly, a good reason to make this system a preferred one for a lot of models is the clear, synthetic description of its dynamics in terms of Israel's junction conditions [1,2] (the null case, considered in detail in the seminal paper of C. Barrabes and W. Israel [3], is also interesting for null-like surfaces [4], i.e. light-like matter shells [5], for which an Hamiltonian treatment is given in [6] generalizing the approach described in [7]). Using this formalism², which has an intuitive geometric meaning, many relevant aspects of gravitation have been brought into light.

Gravitating shells have indeed been considered as natural models for different astrophysical problems: the description of variable cosmic objects [9] and of specific aspects (like ejection [10] or crossing of layers [11], critical phenomena [12], perturbations [13] and back-reaction [14]) in gravitational collapse [15,16,17] are only a few examples.

Moreover, at larger scales, specific configurations of shells have also been considered to construct cosmological models [18] (even with hierarchical (fractal) structure [19]), to analyze phase transitions in the early universe [20] or to describe cosmological voids [21]; semiclassical models have tackled the problem of avoiding the initial singularity of the Big-Bang scenario by quantum tunnelling [22,23,24].

As a matter of fact quantum semiclassical models have conveniently been employed as useful simple examples to better understand possible properties and modifications to the spacetime structure at scales at which quantum effects should give significant contributions to gravitational physics. Apart from the quantization of the gravitating shell itself (as in [25,26]), considered also in the context of gravitational collapse [27], models have been proposed to study quantum properties of black holes [28,29,30] and their formation process [31] as well as to analyze wormhole spacetimes [32,33,,35] and the quantum stabilization of their instability [36,37], targeting the fuzzy properties of spacetime foam [38] and Planck scale physics [39].

Other problems of fundamental nature in quantum gravity have received attention through the study of shell dynamics: as an exemplificative list, we mention here Hawking radiation [40,30,41] the horizon problem in wormhole spacetimes [42], the time problem in canonical relativity [43], the problem of localization of gravitational energy [44], the thermodynamics of self-gravitating systems [45,46] and the possibility of connecting compact with non-compact dimensions [47].
Many of the above discussions have been performed under the simplifying assumption of spherical symmetry: this is especially useful in the quantum treatment, because the minisuperspace approximation greatly reduces the complexity of the mathematical treatment.
But, at least at the classical level, studies have also been performed for cylindrical models (see e.g. [48,49,50]).

With the development of the models shortly cited above, particularly those involved with the quantization of the system, many subtleties emerged as byproducts of corresponding difficulties already encountered in tentative approaches to Quantum Gravity and mainly related to the reparametrization invariance of the theory. Since the junction conditions, essentially, are a first integral of the equations of motion of the shell, many authors revolved their attention to the derivation of these equations starting from an action principle. A consistent Lagrangian/Hamiltonian formalism has been developed [51,52,,54] (also reduced by spherical symmetry [55]), and the relevant degrees of freedom of the system [56] discussed together with a variational principle, which is also the subject of [57,58] (interesting considerations can also be found in [59]).

Recently even more interest in the thin shell formalism is coming thanks to the development of brane world scenarios, where our universe is seen as a four dimensional brane embedded in a five dimensional space [60,61]. This configuration can be given a wormhole interpretation [62] and has also been analyzed from the point of view of energy conditions [63] (not) satisfied in the higher dimensional background.

In these and other studies, different cases of junctions between spacetimes have been considered: for example between anti de Sitter and anti de Sitter [62], Friedmann-Robertson-Walker and Friedmann-Robertson-Walker [34], Minkowski and Minkowski [37,38], Schwarzschild and Schwarzschild [36,11,9,6], Reißner-Nordström and Reißner-Nordström [15,33], de Sitter and Reißner-Nordström [63], de Sitter and Schwarzschild [64,23,81], de Sitter and Schwarzschild-de Sitter [65], de Sitter and Vaidya [24], Friedmann-like and Reißner-Nordström [66], Minkowski and Friedmann [21], Minkowski and Reißner-Nordström [15,41,67], Minkowski and Schwarzschild [28,31,25,14,43,29], Minkowski and Vaidya [12], Schwarzschild and Reißner-Nordström [67], Schwarzschild and Schwarzschild-anti de Sitter [47], Schwarzschild and Vaidya [46], Tolman and Friedman [19], Lemaître-Tolman-Bondi and Lemaître-Tolman-Bondi [18].

In this paper we are also going to use a general relativistic shell to analyze, even if only at the semiclassical level, the problem of quantization of a gravitational system. We will restrict ourselves, as it has been done in many of the papers cited above, to the spherically symmetric case and we will study the semiclassical quantum dynamics in the case in which the shell separates an interior spacetime of the de Sitter geometry, from an exterior of the extremal Reißner-Nordström type. An observer crossing the shell will naively see some non-vanishing vacuum energy density to be converted into physical properties like charge and mass. From the classical dynamics there are no restrictions on the values of the physical parameters characterizing the geometry of spacetime. But, starting from a Hamiltonian description of the shell dynamics, we will try to analyze its quantum behaviour. Lacking a full theory of quantum gravity, which would of course be the natural setting for this kind of problem, we will tackle it only at the semiclassical level: under this word, we will understand that the action for the shell is given as an integer multiple of the quantum, $\hbar$. We will see that this condition results in a constraint on the parameters for the interior and exterior geometries. This is hardly surprising: indeed a full quantum theory of gravity, would have the task of determining the probability amplitude for a given configuration of the three-geometries taught as points in superspace; in our quantum minisuperspace approach, the only free parameters remain the constants (de Sitter cosmological horizon, charge and mass) fixing the interior and exterior metrics, and is thus as a relation among them that the semiclassical quantization conditions realizes itself.

With the above ideas in mind the paper is organized as follows.
In section 2 we will set up our model by giving all relevant definitions; we will also recall some well known results adapted to our special case, to fix notations and conventions, and will present all relevant dynamical quantities for the computations that follows. The Bohr-Sommerfeld quantization condition is also recalled. Then, in section 3 the classical dynamics of the system is sketched and the associated spacetime structure discussed, with particular emphasis on the bounded trajectories. This prepares the ground for section 4, where the classical action is numerically evaluated for bounded trajectories. This result is then used in section 5 to show how the Bohr-Sommerfeld quantization condition characterizes the properties of the semiclassical quantum system. After a preliminary rough estimate (subsection 5.1), we present the semi-classical results for the quantum levels of the shell and the corresponding internal/external geometries and approximate the results with a properly chosen analytic (polynomial) expression. Discussion about the results and possible refinements of the model follow in section 6.
Five short appendices are devoted to a more detailed analysis of some technical points. The turning points of the classical motion are discussed in A. The issue about the stability of the classical solution against single particle decay is studied in B. C shows that the bounded trajectories are not affected by change of direction of the normal to the shell trajectory. The characterization of the singularity that appears in the integral for the computation of the classical action as an integrable one is done in D and the determination of the leading terms in the integrand of the same computation is the topic of E.